Skill: Identify the minor associated with a matrix element

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The minor associated with an entry of a square matrix is the determinant of the submatrix formed by deleting the entry’s entire row and column; for a $3\times3$ matrix, this is the determinant of a $2\times2$ submatrix. A minor depends on the entry’s position but does not include the alternating sign used to form its cofactor; broader higher-order determinant theory is not included.

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Detailed Explanation: Identify the minor associated with a matrix element

To find the minor associated with a matrix entry:

Locate the entry and note its row and column.
Delete that entire row and column.
Find the determinant of the remaining submatrix.

For example, find the minor associated with the entry $5$ in

A=\begin{pmatrix} 2 & 5 & 1\\ 3 & 4 & 6\\ 7 & 0 & 8 \end{pmatrix}.

The entry $5$ is in row $1$ , column $2$ . Delete row $1$ and column $2$ :

\begin{pmatrix} 3 & 6\\ 7 & 8 \end{pmatrix}.

Its determinant is

(3)(8)-(6)(7)=24-42=-18.

So the minor associated with the entry $5$ is $\boxed{-18}$ . A minor is just this determinant; it does not include any extra alternating sign.

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Matrices - Matrix of Minors (3x3) - Single Value

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