Minor

Definition

Let $K$ be a commutative ring, let $A$ be an $m \times n$ matrix over $K$, and let $1 \le k \le \min(m, n)$. A minor of order $k$ of $A$ is the determinant of a $k \times k$ submatrix of $A$. Such a submatrix consists of the entries in $k$ chosen rows and $k$ chosen columns of $A$. For an $n \times n$ matrix $A$, delete row $i$ and column $j$ to get an $(n-1) \times (n-1)$ matrix. The minor $M_{ij}$ is the determinant of this matrix.

Remarks

The minors $M_{ij}$ are called first minors. Over a field, the rank of $A$ is the largest $k$ such that $A$ has a nonzero minor of order $k$. The minors with the same chosen rows and columns are principal minors.

Explanations

L1 human

A minor is the determinant of a smaller square piece of a matrix. For a $3 \times 3$ matrix, we delete one row and one column and take the $2 \times 2$ determinant of what is left. Minors appear in the expansion of determinants and in the formula for the inverse.