Matrix Minors
This page follows the JoeCMath video Master Matrix Minors in 3 Minutes. The video defines the minor $M_{ij}$, then works through symbolic and numeric examples for a $3\times 3$ matrix.
Main Idea
The video treats a minor as a smaller determinant, not as a signed cofactor.
For a square matrix
\[A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix},\]the minor of entry $a_{ij}$ is denoted $M_{ij}$. To find $M_{ij}$, delete row $i$ and column $j$, then take the determinant of what remains.
The index on a minor tells you which row and column to remove before taking the determinant.
Watch this section: minor of a square matrix at 0:00.
Finding $M_{11}$
The first worked setup in the video is $M_{11}$.
Start with
\[A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}.\]The subscript $11$ 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.
The Remaining Minors
The video then moves through the other minors for a $3\times 3$ matrix. Each one uses the same instruction: delete the row and column named by the subscript.
$M_{12}$: delete row $1$ and column $2$.
\[\begin{aligned} M_{12} &=\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix}\\ &=a_{21}a_{33}-a_{23}a_{31} \end{aligned}\]$M_{13}$: delete row $1$ and column $3$.
\[\begin{aligned} M_{13} &=\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}\\ &=a_{21}a_{32}-a_{22}a_{31} \end{aligned}\]$M_{21}$: delete row $2$ and column $1$.
\[\begin{aligned} M_{21} &=\begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{vmatrix}\\ &=a_{12}a_{33}-a_{13}a_{32} \end{aligned}\]$M_{22}$: delete row $2$ and column $2$.
\[\begin{aligned} M_{22} &=\begin{vmatrix} a_{11} & a_{13} \\ a_{31} & a_{33} \end{vmatrix}\\ &=a_{11}a_{33}-a_{13}a_{31} \end{aligned}\]$M_{23}$: delete row $2$ and column $3$.
\[\begin{aligned} M_{23} &=\begin{vmatrix} a_{11} & a_{12} \\ a_{31} & a_{32} \end{vmatrix}\\ &=a_{11}a_{32}-a_{12}a_{31} \end{aligned}\]$M_{31}$: delete row $3$ and column $1$.
\[\begin{aligned} M_{31} &=\begin{vmatrix} a_{12} & a_{13} \\ a_{22} & a_{23} \end{vmatrix}\\ &=a_{12}a_{23}-a_{13}a_{22} \end{aligned}\]$M_{32}$: delete row $3$ and column $2$.
\[\begin{aligned} M_{32} &=\begin{vmatrix} a_{11} & a_{13} \\ a_{21} & a_{23} \end{vmatrix}\\ &=a_{11}a_{23}-a_{13}a_{21} \end{aligned}\]$M_{33}$: delete row $3$ and column $3$.
\[\begin{aligned} M_{33} &=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}\\ &=a_{11}a_{22}-a_{12}a_{21} \end{aligned}\]This section is less about memorizing all eight formulas and more about seeing the row-column deletion pattern over and over.
Watch this section: all eight remaining minors at 1:06.
Numeric Example: $M_{22}$
The first numeric matrix in the video is
\[A= \begin{bmatrix} 1 & 7 & 5 \\ 4 & 3 & 8 \\ 6 & 9 & 2 \end{bmatrix}.\]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 $ad-bc$,
\[M_{22}=1\cdot 2-6\cdot 5=2-30=-28.\]Deleting row $2$ and column $2$ leaves the corner entries from rows $1$ and $3$.
Watch this section: find $M_{22}$ of a $3\times 3$ matrix at 1:52.
Numeric Example: $M_{31}$
The second numeric example uses the same matrix:
\[A= \begin{bmatrix} 1 & 7 & 5 \\ 4 & 3 & 8 \\ 6 & 9 & 2 \end{bmatrix}.\]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
\[M_{31}=7\cdot 8-3\cdot 5=56-15=41.\]This example is a good reminder that the entry $a_{31}$ is not the value of the minor. It tells you what row and column to remove.
Watch this section: find $M_{31}$ of a $3\times 3$ matrix at 2:23.
Timestamp Guide
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.