Matrix Cofactors
This video goes over how to find cofactors within a given matrix.
Cofactor Formula
To find the cofactor for entry $a_{ij}$, we can use the following formula
\[C_{ij}=(-1)^{i+j}M_{ij}.\]$M_{ij}$ is the minor of $a_{ij}$: we find this by removing row $i$ and column $j$ and taking the determinant of what remains.
$(-1)^{i+j}$ is then $(+1)$ if $i+j$ is even, or $(-1)$ if $i+j$ is odd.
NOTE: $i+j$ is the sum of the row and column indices associated with entry $a_{ij}$.
Watch this section: cofactor definition at 0:00.
The Pattern of $(-1)^{i+j}$ (AKA the Sign Chart)
Lets create a pattern to remember what $(-1)^{i+j}$ looks like for any square matrix.
If $i+j$ is even, then
\[(-1)^{i+j}=1.\]If $i+j$ is odd, then
\[(-1)^{i+j}=-1.\]That creates the checkerboard sign pattern
\[\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}.\]The top-left position will always be $(+)$ since the row and column index is $1$, so adding $1+1=2$ and $(-1)^2=1$.
More examples:
- Position $(1,3)$ has $1+3=4$, so it keeps a positive sign.
- Position $(1,2)$ has $1+2=3$, so it gets a negative sign.
Watch this section: potential values of $(-1)^{i+j}$ at 0:20 and sign chart summary at 1:03.
Finding $C_{11}$
The video then starts with the matrix
\[A= \begin{bmatrix} 1 & 7 & 5 \\ 4 & 3 & 8 \\ 6 & 9 & 2 \end{bmatrix}\]and asks for the cofactor of $a_{11}$.
Since $1+1=2$ is even, the sign is positive:
\[(-1)^{1+1}=1.\]Now find the minor $M_{11}$ by deleting row $1$ and column $1$:
\[M_{11}=\begin{vmatrix} 3 & 8 \\ 9 & 2 \end{vmatrix}.\]Using the $2\times 2$ determinant rule,
\[M_{11}=3\cdot 2-8\cdot 9=6-72=-66.\]Therefore
\[C_{11}=(-1)^{1+1}M_{11}=1(-66)=-66.\]The cofactor did not change the sign here because $C_{11}$ sits in a positive position on the sign chart.
Watch this section: quick minor calculation at 1:41 and finding $C_{11}$ at 2:21.
The Minor Matrix
After finding the first cofactor, the video fills in the minor matrix for the same $3\times 3$ matrix. Each entry in this matrix is a minor value before the cofactor sign pattern is applied.
For the matrix $A$, the minor matrix shown in the video is
\[M= \begin{bmatrix} -66 & -40 & 18 \\ -31 & -28 & -33 \\ 41 & -12 & -25 \end{bmatrix}.\]NOTE: Each position of $\textbf{M}$ is $M_{ij}$ is the minor of $a_{ij}$.
Watch this section: populating the minor matrix at 2:43.
From Minors To Cofactors (Applying the Sign Chart to the Minor Matrix)
For this video, the sign chart helps us move from the minor matrix to the cofactor matrix.
Start with
\[M= \begin{bmatrix} -66 & -40 & 18 \\ -31 & -28 & -33 \\ 41 & -12 & -25 \end{bmatrix}\]and apply
\[\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}.\]For row $1$, the signs are $+$, $-$, $+$:
\[C_{11}=-66,\qquad C_{12}=40,\qquad C_{13}=18.\]For row $2$, the signs are $-$, $+$, $-$:
\[C_{21}=31,\qquad C_{22}=-28,\qquad C_{23}=33.\]For row $3$, the signs are $+$, $-$, $+$:
\[C_{31}=41,\qquad C_{32}=12,\qquad C_{33}=-25.\]So the cofactor matrix is
\[C= \begin{bmatrix} -66 & 40 & 18 \\ 31 & -28 & 33 \\ 41 & 12 & -25 \end{bmatrix}.\]Watch this section: finding $C_{11}$ at 3:01, finding $C_{12}$ at 3:19, and finishing the remaining cofactors at 3:44.
Minor Matrix Compared With Cofactor Matrix
Near the end, the video places the minor matrix, sign chart, and cofactor matrix side by side.
The minor matrix is
\[M= \begin{bmatrix} -66 & -40 & 18 \\ -31 & -28 & -33 \\ 41 & -12 & -25 \end{bmatrix}.\]The cofactor matrix is
\[C= \begin{bmatrix} -66 & 40 & 18 \\ 31 & -28 & 33 \\ 41 & 12 & -25 \end{bmatrix}.\]The cofactor matrix has the same positions as the minor matrix, but some entries change sign.
Watch this section: minor matrix, sign chart, and cofactor matrix at 4:14 and comparison at 4:27.
Timestamp Guide
Related Linear Algebra Work
If the minor step feels fast, review the matrix minors companion guide. To see cofactors used inside determinant calculations, use the cofactor expansion companion guide.
You can also browse determinant notes or the JoeCMath linear algebra playlist: watch the playlist on YouTube.