Matrix Minors

Simple overview of matrix minors.

Main Idea

For a given matrix $A$

\[A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\]

we find the the minor of entry $a_{ij}$ by deleting row $i$ and column $j$ and taking the determinant of what remains.

The minor of $a_{ij}$ is denoted $M_{ij}$.

Watch this section: minor of a square matrix at 0:00.

Finding $M_{11}$

Consider

\[A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\]

What is $M_{11}$? (This is the minor of $a_11$)

The subscript of $M_{11}$, the $11$, means delete row $1$ and column $1$.

After deleting that row and column, the remaining matrix is

\[\begin{bmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{bmatrix}\]

So

\[M_{11}=\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix}\]

Using the $2\times 2$ determinant rule

\[\begin{vmatrix} a & b \\ c & d \end{vmatrix}=ad-bc\]

we get

\[M_{11}=a_{22}a_{33}-a_{23}a_{32}.\]

Watch this section: finding the minor of entry $a_{11}$ at 0:17.

Numeric Example: $M_{22}$

Consider

\[A= \begin{bmatrix} 1 & 7 & 5 \\ 4 & 3 & 8 \\ 6 & 9 & 2 \end{bmatrix}\]

What is $M_{22}$?

To find $M_{22}$, delete row $2$ and column $2$.

That leaves

\[\begin{bmatrix} 1 & 5 \\ 6 & 2 \end{bmatrix}\]

Now take the determinant

\[M_{22}=\begin{vmatrix} 1 & 5 \\ 6 & 2 \end{vmatrix}\]

Using $ad-bc$

\[M_{22}=1\cdot 2-6\cdot 5=2-30=-28.\]

Watch this section: find $M_{22}$ of a $3\times 3$ matrix at 1:52.

Numeric Example: $M_{31}$

Using the same matrix

\[A= \begin{bmatrix} 1 & 7 & 5 \\ 4 & 3 & 8 \\ 6 & 9 & 2 \end{bmatrix}\]

What is $M_{31}$?

To find $M_{31}$, delete row $3$ and column $1$.

That leaves

\[\begin{bmatrix} 7 & 5 \\ 3 & 8 \end{bmatrix}\]

So

\[M_{31}=\begin{vmatrix} 7 & 5 \\ 3 & 8 \end{vmatrix}\]

Then

\[M_{31}=7\cdot 8-3\cdot 5=56-15=41.\]

Watch this section: find $M_{31}$ of a $3\times 3$ matrix at 2:23.

Timestamp Guide

Section	What is shown	Video
Minor definition	Define $M_{ij}$ by deleting row $i$ and column $j$.	0:00
$M_{11}$ setup	Delete row $1$ and column $1$ from a symbolic $3\times 3$ matrix.	0:17
Remaining minors	Work through the other symbolic minors.	1:06
Numeric $M_{22}$	Delete row $2$ and column $2$, then compute the $2\times 2$ determinant.	1:52
Numeric $M_{31}$	Delete row $3$ and column $1$, then compute the $2\times 2$ determinant.	2:23

Related Linear Algebra Work

Minors are one piece of cofactor expansion. For the next step, use the cofactor expansion companion guide.

You can also browse determinant notes or the JoeCMath linear algebra playlist: watch the playlist on YouTube.