Matrix Cofactors
This page follows the JoeCMath video Master Matrix Cofactors in 5 Minutes. The video builds from the cofactor formula, reviews the sign pattern, and then uses a $3\times 3$ matrix to turn a minor matrix into a cofactor matrix.
Cofactor Rule
The video starts by defining a cofactor as a signed minor. For entry $a_{ij}$, the cofactor is
\[C_{ij}=(-1)^{i+j}M_{ij}.\]The $M_{ij}$ part is the minor: delete row $i$ and column $j$, then take the determinant of what remains. The $(-1)^{i+j}$ part decides whether that minor keeps its sign or changes sign.
This video uses cofactors as signed minors, with the sign chosen by the row-column position.
Watch this section: cofactor definition at 0:00.
The Sign Pattern
The next part of the video focuses on the factor $(-1)^{i+j}$.
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
\[\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}.\]For example, 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
\[M= \begin{bmatrix} -66 & -40 & 18 \\ -31 & -28 & -33 \\ 41 & -12 & -25 \end{bmatrix}.\]This matrix records the determinant values after the row-column deletion step. It is not yet the cofactor matrix because the checkerboard signs have not all been applied.
Watch this section: populating the minor matrix at 2:43.
From Minors To Cofactors
For this video, the sign chart is the bridge between the minor matrix and 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. The positive positions stay the same. The negative positions flip.
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.