\begin{definition}[Definition:Matrix of Minors]
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $a_{i j}$ denote the element whose indices are $\tuple {i, j}$:
:$\mathbf A = \begin {pmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\
 \vdots &  \vdots & \ddots &  \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n} \\
\end {pmatrix}$
For each $a_{i j}$, let $b_{i j}$ denote the minor of $a_{i j}$:
:$b_{i j} = \begin {vmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1, i - 1} & a_{1, i + 1} & \cdots & \cdots & a_{1 n} \\
a_{2 1} & a_{2 2} & \cdots & a_{2, i - 1} & a_{2, i + 1} & \cdots & \cdots & a_{2 n} \\
 \vdots &  \vdots & \ddots & \vdots &  \vdots & \ddots & \vdots \\
a_{j - 1, 1} & a_{j - 1, 2} & \cdots & a_{j - 1, i - 1} & a_{j - 1, i + 1} & \cdots & \cdots & a_{j - 1, n} \\
a_{j + 1, 1} & a_{j + 1, 2} & \cdots & a_{j + 1, i - 1} & a_{j + 1, i + 1} & \cdots & \cdots & a_{j + 1, n} \\
 \vdots &  \vdots & \ddots & \vdots &  \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n, i - 1} & a_{n, i + 1} & \cdots & \cdots & a_{n n} \\
\end {vmatrix}$
Then $\mathbf B = \sqbrk b_n$ is called the '''matrix of minors''' of $\mathbf A$.
:$\mathbf B = \begin {pmatrix}
b_{1 1} & b_{1 2} & \cdots & b_{1 n} \\
b_{2 1} & b_{2 2} & \cdots & b_{2 n} \\
 \vdots &  \vdots & \ddots &  \vdots \\
b_{n 1} & b_{n 2} & \cdots & b_{n n} \\
\end {pmatrix}$
\end{definition}