Definition:Minor of Determinant/Notation/Order n-1

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Definition

The conventional notation for the minor of a determinant is cumbersome for a minor of order $n-1$.


Let $\mathbf A = \left[{a}\right]_n$ be a square matrix of order $n$.

Let $D := \det \left({\mathbf A}\right)$ denote the determinant of $\mathbf A$.


Let a submatrix $\mathbf B$ of $\mathbf A$ be of order $n - 1$.

Let:

$j$ be the row of $\mathbf A$ which is not included in $\mathbf B$
$k$ be the column of $\mathbf A$ which is not included in $\mathbf B$.

Thus, let $\mathbf B := \mathbf A \left({j; k}\right)$.

Then $\det \left({B}\right)$ can be denoted:

$D_{i j}$


That is, $D_{ij}$ is the minor of order $n-1$ obtained from $D$ by deleting all the elements of row $i$ and column $j$.