A method to compute the determinant of square matrices of order five and six

1. Introduction

Let \(A\) be a \(n\mathrm{\ \times \ }n\) matrix: \[\left[A_{n\mathrm{\ \times \ }n}\right]\mathrm{=}\left[ \begin{array}{cc} \begin{array}{cc} a_{\mathrm{11}} & a_{\mathrm{12}} \\ a_{\mathrm{21}} & a_{\mathrm{22}} \end{array} & \begin{array}{cc} \mathrm{\cdots } & a_{\mathrm{1}n} \\ \mathrm{\cdots } & a_{\mathrm{2}n} \end{array} \\ \begin{array}{cc} \mathrm{\vdots } & \mathrm{\vdots } \\ a_{n\mathrm{1}} & a_{n\mathrm{2}} \end{array} & \begin{array}{cc} \mathrm{\ddots } & \mathrm{\vdots } \\ \mathrm{\cdots } & a_{nn} \end{array} \end{array} \right].\]

Definition 1. The determinant of the matrix of order \(n\mathrm\times\mathrm{\ }n\) is the sum \[\left|A_{n\mathrm{\times }n}\right|\mathrm{=}\left| \begin{array}{cc} \begin{array}{cc} a_{\mathrm{11}} & a_{\mathrm{12}} \\ a_{\mathrm{21}} & a_{\mathrm{22}} \end{array} & \begin{array}{cc} \mathrm{\cdots } & a_{\mathrm{1}n} \\ \mathrm{\cdots } & a_{\mathrm{2}n} \end{array} \\ \begin{array}{cc} \mathrm{\vdots } & \mathrm{\vdots } \\ a_{n\mathrm{1}} & a_{n\mathrm{2}} \end{array} & \begin{array}{cc} \mathrm{\ddots } & \mathrm{\vdots } \\ \mathrm{\cdots } & a_{nn} \end{array} \end{array} \right|\mathrm{=}\sum_{S_n}{{\varepsilon }_{j_{\mathrm{1}}j_{\mathrm{2}}\mathrm{\dots }j_n}\mathrm{\cdot }a_{j_{\mathrm{1}}}\mathrm{\cdot }a_{j_{\mathrm{2}}}\mathrm{\cdot }\mathrm{\dots }\mathrm{\ }{\mathrm{\cdot }a}_{j_n},}\] ranging over the symmetric permutation group \(S_n\), where \[{\varepsilon }_{j_{\mathrm{1}}j_{\mathrm{2}}\mathrm{\dots }j_n}\mathrm{=}\left\{ \begin{array}{c} \mathrm{\ \ +1,\ if\ \ }j_{\mathrm{1}}j_{\mathrm{2}}\mathrm{\dots }j_n\mathrm{,\ is\ an\ even\ permutation} \\ \mathrm{-}\mathrm{1,\ if\ }{\mathrm{\ }j}_{\mathrm{1}}j_{\mathrm{2}}\mathrm{\dots }j_n\mathrm{,\ is\ an\ odd\ permutation.} \end{array} \right.\]

Definition 2. [1] Let \(A_2={\left[ \begin{array}{cc} a_{11} & a_{22} \\ a_{21} & a_{22} \end{array} \right]}_{2\times 2}\) and \(B_2={\left[ \begin{array}{cc} b_{11} & b_{22} \\ b_{21} & b_{22} \end{array} \right]}_{2\times 2}\) are two real matrices of order \(2\times2\). If \(\left|B_2\right|\neq 0\) and \(b_{ij}\neq 0,\ \left(\forall i,j=1,2\right)\), then the duplex fraction or duplex division of the determinant of \(\left|A_2\right|\) on \(\left|B_2\right|\) is defined as follows \[\frac{\left|A_2\right|}{\overline{\left|B_2\right|}}=\frac{\underline{\left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right|}}{\left| \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right|}=\frac{\left| \begin{array}{cc} \frac{a_{11}}{b_{11}} & \frac{a_{12}}{b_{12}} \\ \frac{a_{21}}{b_{21}} & \frac{a_{22}}{b_{22}} \end{array} \right|}{\left| \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right|}.\]

Definition 3. [2] Let \(A_3={\left[ \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right]}_{3\times 3}\) and \(B_3={\left[ \begin{array}{ccc} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{array} \right]}_{3\times 3}\) are two matrices of order \(3\times 3\) such that \(b_{ij}\neq 0,\ \left(\forall i,j=1,2,3\right)\) and \(B_3\) is doubly nonsingular, then the star fraction of \(A_3\) on \(B_3\) is defined as \[{\left(\frac{A_3}{B_3}\right)}^*={\left(\frac{{\left[ \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right]}_{3\times 3}}{{\left[ \begin{array}{ccc} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{array} \right]}_{3\times 3}}\right)=\frac{\left| \begin{array}{cc} \frac{\left| \begin{array}{cc} \frac{a_{11}}{b_{11}} & \frac{a_{12}}{b_{12}} \\ \frac{a_{21}}{b_{21}} & \frac{a_{22}}{b_{22}} \end{array} \right|}{\left| \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right|} & \frac{\left| \begin{array}{cc} \frac{a_{12}}{b_{12}} & \frac{a_{13}}{b_{13}} \\ \frac{a_{22}}{b_{22}} & \frac{a_{23}}{b_{23}} \end{array} \right|}{\left| \begin{array}{cc} b_{12} & b_{13} \\ b_{22} & b_{23} \end{array} \right|} \\ \frac{\left| \begin{array}{cc} \frac{a_{21}}{b_{21}} & \frac{a_{22}}{b_{22}} \\ \frac{a_{31}}{b_{31}} & \frac{a_{32}}{b_{32}} \end{array} \right|}{\left| \begin{array}{cc} b_{21} & b_{122} \\ b_{31} & b_{32} \end{array} \right|} & \frac{\left| \begin{array}{cc} \frac{a_{22}}{b_{22}} & \frac{a_{23}}{b_{23}} \\ \frac{a_{32}}{b_{32}} & \frac{a_{33}}{b_{33}} \end{array} \right|}{\left| \begin{array}{cc} b_{22} & b_{23} \\ b_{32} & b_{33} \end{array} \right|} \end{array} \right|}{\left| \begin{array}{ccc} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{array} \right|}}^*.\]

Definition 4. [3] Let \(B_n={[b_{ij}]}_{n\times n}\) be a square real matrix of order \(n\), then the Dodgson’s condensation of matrix \(B_n\) is a \((n-1\ )\times (\ n-1)\) matrix defined as: \[DC\left(B_n\right)={\left[ \begin{array}{ccc} \left| \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right| & \cdots & \left| \begin{array}{cc} b_{1(n-1)} & b_{1n} \\ b_{2(n-1)} & b_{2n} \end{array} \right| \\ \vdots & \ddots & \vdots \\ \left| \begin{array}{cc} b_{(n-1)1} & b_{(n-1)2} \\ b_{n1} & b_{n2} \end{array} \right| & \cdots & \left| \begin{array}{cc} b_{(n-1)(n-1)} & b_{(n-1)n} \\ b_{n(n-1)} & b_{nn} \end{array} \right| \end{array} \right]}_{(n-1)\times (n-1)}.\]

Definition 5. Let \(A_4={\left[ \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array} \right]}_{4\times 4}\) be a square matrix of order 4, then the twice Dodgsons’s condensation is defined as \[DC\left(A_4\right)={\left[ \begin{array}{ccc} \left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| & \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| & \left| \begin{array}{cc} a_{13} & a_{14} \\ a_{23} & a_{24} \end{array} \right| \\ \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \\ \left| \begin{array}{cc} a_{31} & a_{32} \\ a_{41} & a_{42} \end{array} \right| & \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \end{array} \right]}_{3\times 3},\] and \[DC\left(DC\left(A_4\right)\right)={\left[ \begin{array}{cc} \left| \begin{array}{cc} \left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| & \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| \\ \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \end{array} \right| & \left| \begin{array}{cc} \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| & \left| \begin{array}{cc} a_{13} & a_{14} \\ a_{23} & a_{24} \end{array} \right| \\ \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \end{array} \right| \\ \left| \begin{array}{cc} \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \\ \left| \begin{array}{cc} a_{31} & a_{32} \\ a_{41} & a_{42} \end{array} \right| & \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| \end{array} \right| & \left| \begin{array}{cc} \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \\ \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \end{array} \right| \end{array} \right]}_{2\times 2}.\] The twice Dodgson’s condensation Duplex fraction for the square matrix of order 4 is defined as follows: \[FDC\left(DC\left(A_4\right)\right)=\left[ \begin{array}{cc} \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| & \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| \\ \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \end{array} \right|}{a_{22}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| & \left| \begin{array}{cc} a_{13} & a_{14} \\ a_{23} & a_{24} \end{array} \right| \\ \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \end{array} \right|}{a_{23}} \\ \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \\ \left| \begin{array}{cc} a_{31} & a_{32} \\ a_{41} & a_{42} \end{array} \right| & \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| \end{array} \right|}{a_{32}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \\ \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \end{array} \right|}{a_{33}} \end{array} \right]\ .\]

Definition 6. Let \(A_5={\left[ \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right]}_{5\times 5}\) be a square matrix of order 5, then the thrice Dodgsons’s condensation is defined as follows: \[DC\left(A_5\right)={\left[ \begin{array}{cccc} \left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| & \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| & \left| \begin{array}{cc} a_{13} & a_{14} \\ a_{23} & a_{24} \end{array} \right| & \left| \begin{array}{cc} a_{14} & a_{15} \\ a_{24} & a_{25} \end{array} \right| \\ \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| & \left| \begin{array}{cc} a_{24} & a_{25} \\ a_{34} & a_{35} \end{array} \right| \\ \left| \begin{array}{cc} a_{31} & a_{32} \\ a_{41} & a_{42} \end{array} \right| & \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| & \left| \begin{array}{cc} a_{34} & a_{35} \\ a_{44} & a_{45} \end{array} \right| \\ \left| \begin{array}{cc} a_{41} & a_{42} \\ a_{51} & a_{52} \end{array} \right| & \left| \begin{array}{cc} a_{42} & a_{43} \\ a_{52} & a_{53} \end{array} \right| & \left| \begin{array}{cc} a_{43} & a_{44} \\ a_{53} & a_{54} \end{array} \right| & \left| \begin{array}{cc} a_{44} & a_{45} \\ a_{54} & a_{55} \end{array} \right| \end{array} \right]}_{4\times 4},\] and,\\ \(FDC\left(DC\left(A_5\right)\right)=\) \[{\left[ \begin{array}{ccc} \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| & \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| \\ \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \end{array} \right|}{a_{22}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| & \left| \begin{array}{cc} a_{13} & a_{14} \\ a_{23} & a_{24} \end{array} \right| \\ \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \end{array} \right|}{a_{23}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{13} & a_{14} \\ a_{23} & a_{24} \end{array} \right| & \left| \begin{array}{cc} a_{14} & a_{15} \\ a_{24} & a_{25} \end{array} \right| \\ \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| & \left| \begin{array}{cc} a_{24} & a_{25} \\ a_{34} & a_{35} \end{array} \right| \end{array} \right|}{a_{24}} \\ \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \\ \left| \begin{array}{cc} a_{31} & a_{32} \\ a_{41} & a_{42} \end{array} \right| & \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| \end{array} \right|}{a_{32}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \\ \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \end{array} \right|}{a_{33}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| & \left| \begin{array}{cc} a_{24} & a_{25} \\ a_{34} & a_{35} \end{array} \right| \\ \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| & \left| \begin{array}{cc} a_{34} & a_{35} \\ a_{44} & a_{45} \end{array} \right| \end{array} \right|}{a_{34}} \\ \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{31} & a_{32} \\ a_{41} & a_{42} \end{array} \right| & \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| \\ \left| \begin{array}{cc} a_{41} & a_{42} \\ a_{51} & a_{52} \end{array} \right| & \left| \begin{array}{cc} a_{42} & a_{43} \\ a_{52} & a_{53} \end{array} \right| \end{array} \right|}{a_{42}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \\ \left| \begin{array}{cc} a_{42} & a_{43} \\ a_{52} & a_{53} \end{array} \right| & \left| \begin{array}{cc} a_{43} & a_{44} \\ a_{53} & a_{54} \end{array} \right| \end{array} \right|}{a_{43}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| & \left| \begin{array}{cc} a_{34} & a_{35} \\ a_{44} & a_{45} \end{array} \right| \\ \left| \begin{array}{cc} a_{43} & a_{44} \\ a_{53} & a_{54} \end{array} \right| & \left| \begin{array}{cc} a_{44} & a_{45} \\ a_{54} & a_{55} \end{array} \right| \end{array} \right|}{a_{44}} \end{array} \right]}_{3\times 3},\] The thrice Dodgson’s condensation Duplex fraction for the square matrix of order 5 is defined as follows:
\(FDC\left(DC\left(DC\left(A_5\right)\right)\right)=\) \[{\left[ \begin{array}{cc} \frac{\left| \begin{array}{cc} \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| & \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| \\ \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \end{array} \right|}{a_{22}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| & \left| \begin{array}{cc} a_{13} & a_{14} \\ a_{23} & a_{24} \end{array} \right| \\ \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \end{array} \right|}{a_{23}} \\ \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \\ \left| \begin{array}{cc} a_{31} & a_{32} \\ a_{41} & a_{42} \end{array} \right| & \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| \end{array} \right|}{a_{32}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \\ \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \end{array} \right|}{a_{33}} \end{array} \right|}{\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|} & \frac{\left| \begin{array}{cc} \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| & \left| \begin{array}{cc} a_{13} & a_{14} \\ a_{23} & a_{24} \end{array} \right| \\ \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \end{array} \right|}{a_{23}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{13} & a_{14} \\ a_{23} & a_{24} \end{array} \right| & \left| \begin{array}{cc} a_{14} & a_{15} \\ a_{24} & a_{25} \end{array} \right| \\ \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| & \left| \begin{array}{cc} a_{24} & a_{25} \\ a_{34} & a_{35} \end{array} \right| \end{array} \right|}{a_{24}} \\ \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \\ \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \end{array} \right|}{a_{33}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| & \left| \begin{array}{cc} a_{24} & a_{25} \\ a_{34} & a_{35} \end{array} \right| \\ \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| & \left| \begin{array}{cc} a_{34} & a_{35} \\ a_{44} & a_{45} \end{array} \right| \end{array} \right|}{a_{34}} \end{array} \right|}{\left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right|} \\ \frac{\left| \begin{array}{cc} \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \\ \left| \begin{array}{cc} a_{31} & a_{32} \\ a_{41} & a_{42} \end{array} \right| & \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| \end{array} \right|}{a_{32}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \\ \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \end{array} \right|}{a_{33}} \\ \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{31} & a_{32} \\ a_{41} & a_{42} \end{array} \right| & \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| \\ \left| \begin{array}{cc} a_{41} & a_{42} \\ a_{51} & a_{52} \end{array} \right| & \left| \begin{array}{cc} a_{42} & a_{43} \\ a_{52} & a_{53} \end{array} \right| \end{array} \right|}{a_{42}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \\ \left| \begin{array}{cc} a_{42} & a_{43} \\ a_{52} & a_{53} \end{array} \right| & \left| \begin{array}{cc} a_{43} & a_{44} \\ a_{53} & a_{54} \end{array} \right| \end{array} \right|}{a_{43}} \end{array} \right|}{\left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right|} & \frac{\left| \begin{array}{cc} \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \\ \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \end{array} \right|}{a_{33}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| & \left| \begin{array}{cc} a_{24} & a_{25} \\ a_{34} & a_{35} \end{array} \right| \\ \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| & \left| \begin{array}{cc} a_{34} & a_{35} \\ a_{44} & a_{45} \end{array} \right| \end{array} \right|}{a_{34}} \\ \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \\ \left| \begin{array}{cc} a_{42} & a_{43} \\ a_{52} & a_{53} \end{array} \right| & \left| \begin{array}{cc} a_{43} & a_{44} \\ a_{53} & a_{54} \end{array} \right| \end{array} \right|}{a_{43}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| & \left| \begin{array}{cc} a_{34} & a_{35} \\ a_{44} & a_{45} \end{array} \right| \\ \left| \begin{array}{cc} a_{43} & a_{44} \\ a_{53} & a_{54} \end{array} \right| & \left| \begin{array}{cc} a_{44} & a_{45} \\ a_{54} & a_{55} \end{array} \right| \end{array} \right|}{a_{44}} \end{array} \right|}{\left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right|} \end{array} \right]}_{2\times 2}.\]

2. Some useful Lemmas

To prove our main results we need the following lemmas.

Lemma 7. (Salihu’s method)[4] The determinant of the square matrix \(A_n={\left[ \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array} \right]}_{n\times n}\) is equal to: \[\left| \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array} \right|=\frac{\left| \begin{array}{cc} \left| \begin{array}{ccc} a_{11} & \cdots & a_{1(n-1)} \\ \vdots & \ddots & \vdots \\ a_{(n-1)1} & \cdots & a_{(n-1)(n-1)} \end{array} \right| & \left| \begin{array}{ccc} a_{12} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{(n-1)2} & \cdots & a_{(n-1)n} \end{array} \right| \\ \left| \begin{array}{ccc} a_{21} & \cdots & a_{2(n-1)} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{n(n-1)} \end{array} \right| & \left| \begin{array}{ccc} a_{22} & \cdots & a_{2n} \\ \vdots & \ddots & \vdots \\ a_{n2} & \cdots & a_{nn} \end{array} \right| \end{array} \right|}{\left| \begin{array}{ccc} a_{22} & \cdots & a_{2(n-1)} \\ \vdots & \ddots & \vdots \\ a_{(n-1)2} & \cdots & a_{(n-1)(n-1)} \end{array} \right|}.\]

Lemma 8. [2] Given a square matrix \(A_4={\left[ \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{23} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array} \right]}_{4\times 4}\) of order 4 such that \(\left[ \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right]>0\) and \(\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|\neq 0\). Then \[det\left(A_4\right)=\frac{\underline{\left|DC\left(DC\left(A_4\right)\right)\right|}}{\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|}=\frac{\underline{\left| \begin{array}{cc} \left| \begin{array}{cc} \left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| & \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| \\ \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \end{array} \right| & \left| \begin{array}{cc} \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| & \left| \begin{array}{cc} a_{13} & a_{14} \\ a_{23} & a_{24} \end{array} \right| \\ \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \end{array} \right| \\ \left| \begin{array}{cc} \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \\ \left| \begin{array}{cc} a_{31} & a_{32} \\ a_{41} & a_{42} \end{array} \right| & \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| \end{array} \right| & \left| \begin{array}{cc} \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \\ \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \end{array} \right| \end{array} \right|}}{\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|}\] \[=\frac{\left| \begin{array}{cc} \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| & \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| \\ \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \end{array} \right|}{a_{22}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right| & \left| \begin{array}{cc} a_{13} & a_{14} \\ a_{23} & a_{24} \end{array} \right| \\ \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \end{array} \right|}{a_{23}} \\ \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right| & \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| \\ \left| \begin{array}{cc} a_{31} & a_{32} \\ a_{41} & a_{42} \end{array} \right| & \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| \end{array} \right|}{a_{32}} & \frac{\left| \begin{array}{cc} \left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| & \left| \begin{array}{cc} a_{23} & a_{24} \\ a_{33} & a_{34} \end{array} \right| \\ \left| \begin{array}{cc} a_{32} & a_{33} \\ a_{42} & a_{43} \end{array} \right| & \left| \begin{array}{cc} a_{33} & a_{34} \\ a_{43} & a_{44} \end{array} \right| \end{array} \right|}{a_{33}} \end{array} \right|}{\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|}=\frac{\left|FDC\left(DC\left(A_4\right)\right)\right|}{\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|}.\]

Lemma 9. [2] Consider a square matrix \[A_5={\left[ \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right]}_{5\times 5},\] of order 5, where \(\left[ \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right]\) is a doubly nonsingular matrix with all nonzero elements. Then \[\left|A_5\right|={\left(\frac{DC\left(DC\left(A_5\right)\right)}{\left[ \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right]}\right)}^*=\frac{\underline{\ \ \ \ \ \left|{DC\left(DC\left(DC\left(A_5\right)\right)\right)}_1\right|\ \ \ \ }}{\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44}. \end{array} \right|}\]

Lemma 10. [5] Suppose that \(A\) is a square matrix. Let\(\ B\) be the square matrix obtained from \(A\) by interchanging the location of two rows, or interchanging the location of two columns. Then \(|A|\ =\ -|B|\).

Lemma 11. [5] Suppose that \(A\) is a square matrix. Let \(B\) be the square matrix obtained from \(A\) by multiplying a single row by the scalar \(\alpha \), or by multiplying a single column by the scalar \(\alpha \). Then \(|A|\ =\ \alpha |B|\).

3. Main Results

Theorem 12. Given a square matrix \(A_5={\left[ \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right]}_{5\times 5}\) of order 5, such that \(\left[ \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right]>0\) , \(\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|\neq 0\) and \(\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|\neq 0\), then \[{det \left(A_5\right)\ }=\frac{\left| \begin{array}{cc} \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_1\right]}^*\right| & \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_2\right]}^*\right| \\ \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_3\right]}^*\right| & \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_4\right]}^*\right| \end{array} \right|}{\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|\cdot {\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|}^2}\ .\]

Proof. By Lemma 7, we have: \[\left|A_5\right|=\left| \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right|=\frac{\left| \begin{array}{cc} \left| \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array} \right| & \left| \begin{array}{cccc} a_{12} & a_{13} & a_{14} & a_{15} \\ a_{22} & a_{23} & a_{24} & a_{25} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{42} & a_{43} & a_{44} & a_{45} \end{array} \right| \\ \left| \begin{array}{cccc} a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \\ a_{51} & a_{52} & a_{53} & a_{54} \end{array} \right| & \left| \begin{array}{cccc} a_{22} & a_{23} & a_{24} & a_{25} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{42} & a_{43} & a_{44} & a_{45} \\ a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right| \end{array} \right|}{\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|}\ ,\] by Lemma 10, we have \[\left| \begin{array}{cc} \left| \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array} \right| & \left| \begin{array}{cccc} a_{12} & a_{13} & a_{14} & a_{15} \\ a_{22} & a_{23} & a_{24} & a_{25} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{42} & a_{43} & a_{44} & a_{45} \end{array} \right| \\ \left| \begin{array}{cccc} a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \\ a_{51} & a_{52} & a_{53} & a_{54} \end{array} \right| & \left| \begin{array}{cccc} a_{22} & a_{23} & a_{24} & a_{25} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{42} & a_{43} & a_{44} & a_{45} \\ a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right| \end{array} \right|\] \[=\left| \begin{array}{cc} \ \ \ \ \left| \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array} \right| & -\left| \begin{array}{cccc} a_{15} & a_{12} & a_{13} & a_{14} \\ a_{25} & a_{22} & a_{23} & a_{24} \\ a_{35} & a_{32} & a_{33} & a_{34} \\ a_{45} & a_{42} & a_{43} & a_{44} \end{array} \right| \\ -\left| \begin{array}{cccc} a_{51} & a_{52} & a_{53} & a_{54} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array} \right| & \ \ \ \ \left| \begin{array}{cccc} a_{55} & a_{52} & a_{53} & a_{54} \\ a_{25} & a_{22} & a_{23} & a_{24} \\ a_{35} & a_{32} & a_{33} & a_{34} \\ a_{45} & a_{42} & a_{43} & a_{44} \end{array} \right| \end{array} \right|\ ,\] by Lemma 8, we have: \[\left|A_5\right|=\frac{1}{\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|}\cdot \left| \begin{array}{cc} \frac{\underline{\left|{DC\left(DC\left(A_4\right)\right)}_1\right|}}{\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|} & \frac{\underline{\left|{DC\left(DC\left(A_4\right)\right)}_2\right|}}{\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|} \\ \frac{\underline{\left|{DC\left(DC\left(A_4\right)\right)}_3\right|}}{\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|} & \frac{\underline{\left|{DC\left(DC\left(A_4\right)\right)}_4\right|}}{\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|} \end{array} \right|\ ,\] using Definition 6 and Lemma 11, we obtain \begin{eqnarray*}\left|A_5\right|&=&\frac{1}{\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|}\cdot \frac{1}{{\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|}^2}\cdot \left| \begin{array}{cc} \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_1\right]}^*\right| & \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_2\right]}^*\right| \\ \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_3\right]}^*\right| & \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_4\right]}^*\right| \end{array} \right|\\ &=&\frac{\left| \begin{array}{cc} \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_1\right]}^*\right| & \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_2\right]}^*\right| \\ \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_3\right]}^*\right| & \left|{\left[{FDC\left(DC\left(A_4\right)\right)}_4\right]}^*\right| \end{array} \right|}{\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|\cdot {\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right|}^2}\ .\end{eqnarray*} This complete the prove.

Example 1. Let \(A=\left[ \begin{array}{ccccc} 2 & 1 & 4 & 1 & 3 \\ 1 & 4 & 3 & 2 & 1 \\ 2 & 5 & 2 & 3 & 2 \\ 1 & 1 & 4 & 1 & 2 \\ 2 & 3 & 1 & 4 & 1 \end{array} \right]\) be a square matric of order 5, then by using Theorem 12, we have \begin{eqnarray*}\left|A_5\right|&=&\left| \begin{array}{ccccc} 2 & 1 & 4 & 1 & 3 \\ 1 & 4 & 3 & 2 & 1 \\ 2 & 5 & 2 & 3 & 2 \\ 1 & 1 & 4 & 1 & 2 \\ 2 & 3 & 1 & 4 & 1 \end{array} \right|\\&=&\frac{\left| \begin{array}{cc} \ \ \ \ \left|{\left[ \begin{array}{cccc} 2 & 1 & 4 & 1 \\ 1 & 4 & 3 & 2 \\ 2 & 5 & 2 & 3 \\ 1 & 1 & 4 & 1 \end{array} \right]}^*\right| & \left|{\left[ \begin{array}{cccc} 3 & 1 & 4 & 1 \\ 1 & 4 & 3 & 2 \\ 2 & 5 & 2 & 3 \\ 2 & 1 & 4 & 1 \end{array} \right]}^*\right| \\ \left|{\left[ \begin{array}{cccc} 2 & 3 & 1 & 4 \\ 1 & 4 & 3 & 2 \\ 2 & 5 & 2 & 3 \\ 1 & 1 & 4 & 1 \end{array} \right]}^*\right| & \ \ \ \ \left|{\left[ \begin{array}{cccc} 1 & 3 & 1 & 4 \\ 1 & 4 & 3 & 2 \\ 2 & 5 & 2 & 3 \\ 2 & 1 & 4 & 1 \end{array} \right]}^*\right| \end{array} \right|}{\left| \begin{array}{ccc} 4 & 3 & 2 \\ 5 & 2 & 3 \\ 1 & 4 & 1 \end{array} \right|\cdot {\left| \begin{array}{cc} 4 & 3 \\ 5 & 2 \end{array} \right|}^2}\\&=&\ \frac{\left| \begin{array}{cc} \ \ \ \ \ \ 70 & \ \ \ \ \ 70 \\ -175 & -350 \end{array} \right|}{-10\cdot {\left(-7\right)}^2}\\ &=&\frac{-12\ 250}{-490}=25.\end{eqnarray*}

Theorem 13. Given a square matrix \(A_6={\left[ \begin{array}{cccccc} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \\ a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66} \end{array} \right]}_{6\times 6}\) of order 6 such that \(\left[ \begin{array}{cccc} a_{22} & a_{23} & a_{24} & a_{25} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{42} & a_{43} & a_{44} & a_{45} \\ a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right]>0\), \(\left| \begin{array}{cccc} a_{22} & a_{23} & a_{24} & a_{25} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{42} & a_{43} & a_{44} & a_{45} \\ a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right|\)\(\neq 0\) and \(\left| \begin{array}{ccc} a_{33} & a_{34} & a_{35} \\ a_{43} & a_{44} & a_{45} \\ a_{53} & a_{54} & a_{55} \end{array} \right|\neq 0\), then \[{det \left(A_6\right)\ }=\frac{\left| \begin{array}{cc} \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_1\right]}^*\right| & \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_2\right]}^*\right| \\ \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_3\right]}^*\right| & \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_4\right]}^*\right| \end{array} \right|}{\left| \begin{array}{cccc} a_{22} & a_{23} & a_{24} & a_{25} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{42} & a_{43} & a_{44} & a_{45} \\ a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right|\cdot {\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|}^2}\ .\]

Proof. By Lemma 7, we have \begin{eqnarray*}\left|A_6\right|&=&\left| \begin{array}{cccccc} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \\ a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66} \end{array} \right|\\ &=&\frac{\left| \begin{array}{cc} \left| \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right| & \left| \begin{array}{ccccc} a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \end{array} \right| \\ \left| \begin{array}{ccccc} a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \\ a_{61} & a_{62} & a_{63} & a_{64} & a_{55} \end{array} \right| & \left| \begin{array}{ccccc} a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \\ a_{62} & a_{63} & a_{64} & a_{65} & a_{56} \end{array} \right| \end{array} \right|}{\left| \begin{array}{cccc} a_{22} & a_{23} & a_{24} & a_{25} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{42} & a_{43} & a_{44} & a_{45} \\ a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right|}\ \end{eqnarray*} by Lemma 10, we know that \[\left| \begin{array}{cc} \left| \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right| & \left| \begin{array}{ccccc} a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \end{array} \right| \\ \left| \begin{array}{ccccc} a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \\ a_{61} & a_{62} & a_{63} & a_{64} & a_{55} \end{array} \right| & \left| \begin{array}{ccccc} a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \\ a_{62} & a_{63} & a_{64} & a_{65} & a_{56} \end{array} \right| \end{array} \right|\] \[=\left| \begin{array}{cc} \left| \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right| & \left| \begin{array}{ccccc} a_{16} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{26} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{36} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{46} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{56} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right| \\ \left| \begin{array}{ccccc} a_{61} & a_{62} & a_{63} & a_{64} & a_{55} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right| & \left| \begin{array}{ccccc} a_{66} & a_{62} & a_{63} & a_{64} & a_{55} \\ a_{26} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{36} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{46} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{56} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right| \end{array} \right|\ ,\] using Lemma 9, we have \[\left|A_6\right|=\frac{1}{\left| \begin{array}{cccc} a_{22} & a_{23} & a_{24} & a_{25} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{42} & a_{43} & a_{44} & a_{45} \\ a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right|}\cdot \left| \begin{array}{cc} \frac{\underline{\ \ \ \ \ \left|{DC\left(DC\left(DC\left(A_5\right)\right)\right)}_1\right|\ \ \ \ }}{\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|} & \frac{\underline{\ \ \ \ \ \left|{DC\left(DC\left(DC\left(A_5\right)\right)\right)}_2\right|\ \ \ \ \ }}{\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|} \\ \frac{\ \underline{\ \ \ \ \ \left|{DC\left(DC\left(DC\left(A_5\right)\right)\right)}_3\right|\ \ \ \ }}{\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|} & \frac{\underline{\ \ \ \ \ \left|DC\left(DC\left(DC\left(A_5\right)\right)\right)\right|\ \ \ \ \ }}{\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|} \end{array} \right|\ \] using Definition 6 and Lemma 11, we obtain \begin{eqnarray*}\left|A_6\right|&=&\frac{1}{\left| \begin{array}{cccc} a_{22} & a_{23} & a_{24} & a_{25} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{42} & a_{43} & a_{44} & a_{45} \\ a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right|}\cdot \frac{1}{{\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|}^2}\\&& \cdot \left| \begin{array}{cc} \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_1\right]}^*\right| & \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_2\right]}^*\right| \\ \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_3\right]}^*\right| & \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_4\right]}^*\right| \end{array} \right|\\ &=&\frac{\left| \begin{array}{cc} \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_1\right]}^*\right| & \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_2\right]}^*\right| \\ \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_3\right]}^*\right| & \left|{\left[{FDC\left(DC\left(DC\left(A_5\right)\right)\right)}_4\right]}^*\right| \end{array} \right|}{\left| \begin{array}{cccc} a_{22} & a_{23} & a_{24} & a_{25} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{42} & a_{43} & a_{44} & a_{45} \\ a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right|\cdot {\left| \begin{array}{ccc} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right|}^2} .\end{eqnarray*} This complete the prove.

Example 2. Let \(A\left[ \begin{array}{cccccc} 1 & 4 & 2 & 1 & 3 & 2 \\ 2 & 1 & 1 & 4 & 2 & 3 \\ 2 & 3 & 4 & 1 & 1 & 2 \\ 3 & 1 & 4 & 2 & 1 & 4 \\ 1 & 4 & 2 & 3 & 4 & 3 \\ 4 & 1 & 1 & 2 & 1 & 3 \end{array} \right]\) be a square matrix of order 6, then by using Theorem 13, we have \begin{eqnarray*}\left|A_6\right|&=&\left| \begin{array}{cccccc} 1 & 4 & 2 & 1 & 3 & 2 \\ 2 & 1 & 1 & 4 & 2 & 3 \\ 2 & 3 & 4 & 1 & 1 & 2 \\ 3 & 1 & 4 & 2 & 1 & 4 \\ 1 & 4 & 2 & 3 & 4 & 3 \\ 4 & 1 & 1 & 2 & 1 & 3 \end{array} \right| \frac{\left| \begin{array}{cc} \ \ \ \ \left|{\left[ \begin{array}{ccccc} 1 & 4 & 2 & 1 & 3 \\ 2 & 1 & 1 & 4 & 2 \\ 2 & 3 & 4 & 1 & 1 \\ 3 & 1 & 4 & 2 & 1 \\ 1 & 4 & 2 & 3 & 4 \end{array} \right]}^*\right| & \left|{\left[ \begin{array}{ccccc} 2 & 4 & 2 & 1 & 3 \\ 3 & 1 & 1 & 4 & 2 \\ 2 & 3 & 4 & 1 & 1 \\ 4 & 1 & 4 & 2 & 1 \\ 3 & 4 & 2 & 3 & 4 \end{array} \right]}^*\right| \\ \left|{\left[ \begin{array}{ccccc} 4 & 1 & 1 & 2 & 1 \\ 2 & 1 & 1 & 4 & 2 \\ 2 & 3 & 4 & 1 & 1 \\ 3 & 1 & 4 & 2 & 1 \\ 1 & 4 & 2 & 3 & 4 \end{array} \right]}^*\right| & \ \ \ \ \left|{\left[ \begin{array}{ccccc} 3 & 1 & 1 & 2 & 1 \\ 3 & 1 & 1 & 4 & 2 \\ 2 & 3 & 4 & 1 & 1 \\ 4 & 1 & 4 & 2 & 1 \\ 3 & 4 & 2 & 3 & 4 \end{array} \right]}^*\right| \end{array} \right|}{\left| \begin{array}{cccc} 1 & 1 & 4 & 2 \\ 3 & 4 & 1 & 1 \\ 1 & 4 & 2 & 1 \\ 4 & 2 & 3 & 4 \end{array} \right|\cdot {\left| \begin{array}{ccc} 1 & 1 & 4 \\ 3 & 4 & 1 \\ 1 & 4 & 2 \end{array} \right|}^2}\\&=& \frac{\left| \begin{array}{cc} 1426 & \ \ 899 \\ 4898 & 2635 \end{array} \right|}{56\cdot {31}^2}=\frac{-645\ 792}{53\ 816}=-12.\end{eqnarray*}

Competing Interests

The author(s) do not have any competing interests in the manuscript.