Let $[m_{i,j}]$ be the n $\times$ n matrix. Denote by $A_{i,j}$ the submatrix of M obtained by deleting its i row and j column
containing $a_{i,j}$. Then the $\det{A_{i,j}}$ is called minor of $m_{i,j}$
For example, let M:
\begin{equation} \begin{aligned} M= \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\\ \end{bmatrix} \\ A_{1,1} = \begin{bmatrix} 5 & 6\\ 8 & 9\\ \end{bmatrix} \\ A_{1,3} = \begin{bmatrix} 2 & 3\\ 8 & 9\\ \end{bmatrix} \end{aligned} \end{equation} The minor of $m_{1,1}$ is $\det{(A_{1,1})} = 5\times 9 - 6\times 8$
The minor of $m_{1,3}$ is $\det{(A_{1,3})} = 2\times 9 - 8\times 3$
If we multiply the minor of $m_{i,j}$ by $(-1)^{i+j}$, then we arrive the defintion of cofactor expansion of $A_{i,j}$ of $a_{i,j}$
\[ \det{(A)} = \sum_{j=1}^{n} a_{1,j} (-1)^{1+j} \det{(M_{1,j})} = \sum_{j=1}^{n} a_{1,j} A_{1,j} \] $\mbox{Cofactor of a matrix}$ \[ \mathbf{C_{i,j}} = (-1)^{i+j}\mathbf{A_{i,j}} \] \[ \mathbf{B} = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{bmatrix} \nonumber \] Cofactor Matrix \begin{equation} \begin{aligned} \mathbf{C} &= \begin{pmatrix} +\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \\ \end{array} \right| & -\left| \begin{array}{cc} a_{21} & a_{23} \\ a_{31} & a_{33} \\ \end{array} \right| +\left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \\[5pt] \end{array} \right| \\ -\left| \begin{array}{cc} a_{12} & a_{13} \\[5pt] a_{32} & a_{33} \\ \end{array} \right| & +\left| \begin{array}{cc} a_{11} & a_{31} \\ a_{13} & a_{33} \\ \end{array} \right| -\left| \begin{array}{cc} a_{11} & a_{12} \\ a_{31} & a_{32} \\ \end{array} \right| \\ +\left| \begin{array}{cc} a_{22} & a_{23} \\[5pt] a_{32} & a_{33} \\ \end{array} \right| & -\left| \begin{array}{cc} a_{21} & a_{23} \\[5pt] a_{31} & a_{33} \\ \end{array} \right| +\left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{array} \right| \end{pmatrix} \nonumber \\ \end{aligned} \end{equation} Its adjugate of matrix is tranpose of Cofactor matrix
\begin{equation} \begin{aligned} Adj(A) &= C^{T} \end{aligned} \end{equation}
containing $a_{i,j}$. Then the $\det{A_{i,j}}$ is called minor of $m_{i,j}$
For example, let M:
\begin{equation} \begin{aligned} M= \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\\ \end{bmatrix} \\ A_{1,1} = \begin{bmatrix} 5 & 6\\ 8 & 9\\ \end{bmatrix} \\ A_{1,3} = \begin{bmatrix} 2 & 3\\ 8 & 9\\ \end{bmatrix} \end{aligned} \end{equation} The minor of $m_{1,1}$ is $\det{(A_{1,1})} = 5\times 9 - 6\times 8$
The minor of $m_{1,3}$ is $\det{(A_{1,3})} = 2\times 9 - 8\times 3$
If we multiply the minor of $m_{i,j}$ by $(-1)^{i+j}$, then we arrive the defintion of cofactor expansion of $A_{i,j}$ of $a_{i,j}$
\[ \det{(A)} = \sum_{j=1}^{n} a_{1,j} (-1)^{1+j} \det{(M_{1,j})} = \sum_{j=1}^{n} a_{1,j} A_{1,j} \] $\mbox{Cofactor of a matrix}$ \[ \mathbf{C_{i,j}} = (-1)^{i+j}\mathbf{A_{i,j}} \] \[ \mathbf{B} = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{bmatrix} \nonumber \] Cofactor Matrix \begin{equation} \begin{aligned} \mathbf{C} &= \begin{pmatrix} +\left| \begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \\ \end{array} \right| & -\left| \begin{array}{cc} a_{21} & a_{23} \\ a_{31} & a_{33} \\ \end{array} \right| +\left| \begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \\[5pt] \end{array} \right| \\ -\left| \begin{array}{cc} a_{12} & a_{13} \\[5pt] a_{32} & a_{33} \\ \end{array} \right| & +\left| \begin{array}{cc} a_{11} & a_{31} \\ a_{13} & a_{33} \\ \end{array} \right| -\left| \begin{array}{cc} a_{11} & a_{12} \\ a_{31} & a_{32} \\ \end{array} \right| \\ +\left| \begin{array}{cc} a_{22} & a_{23} \\[5pt] a_{32} & a_{33} \\ \end{array} \right| & -\left| \begin{array}{cc} a_{21} & a_{23} \\[5pt] a_{31} & a_{33} \\ \end{array} \right| +\left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{array} \right| \end{pmatrix} \nonumber \\ \end{aligned} \end{equation} Its adjugate of matrix is tranpose of Cofactor matrix
\begin{equation} \begin{aligned} Adj(A) &= C^{T} \end{aligned} \end{equation}