---
tag: 0SD528P0EK
kind: definition
name: Minor
msc: [15A15]
defines: 0BT9BYWVRY
uses: [0C3CFHDN6R, 0N1MY1T0W9]
sources:
  - url: https://en.wikipedia.org/wiki/Minor_(linear_algebra)
    locator: General definition
---

## 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|00KDZRW4C0]] 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|0ZC9YTXZRY]] 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.