# Definition:Matrix of Minors

## Definition

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 $\mathbf A$.

$\mathbf B = \begin {pmatrix} b_{1 1} & b_{1 2} & \cdots & b_{1 n} \\ b_{2 1} & b_{2 2} & \cdots & b_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n 1} & b_{n 2} & \cdots & b_{n n} \\ \end {pmatrix}$