Minors and cofactors

Intuition

A large determinant can be broken into smaller ones. Choose an entry, cross out its row and its column, and a smaller square matrix is left. Its determinant, with a sign that depends only on the position, is the cofactor of that entry.

Crossing out one row and one column of a crossword leaves a smaller grid that can be solved on its own. A minor is the determinant of that smaller grid.

For an $n\times n$ matrix $A$ , the minor $M_{ij}$ is the determinant of the $(n-1)\times(n-1)$ matrix left when row $i$ and column $j$ are deleted. The cofactor is the minor with a sign: $C_{ij}=(-1)^{i+j}M_{ij}$ . The sign depends only on the position and follows a chessboard pattern, with $+$ in the top left corner.

C_{ij}=(-1)^{i+j}M_{ij}

Reading the definition

The signs $(-1)^{i+j}$ form the pattern $\begin{pmatrix}+&-&+\\-&+&-\\+&-&+\end{pmatrix}$ : where is even and where it is odd.

Expansion along the first row

\det A=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}

Every one of the six terms takes exactly one entry from row 1, so the terms fall into three groups of two. What multiplies $a_{11}$ in its group is the $2\times2$ determinant left when row 1 and column 1 are deleted, and likewise for the others; the middle group comes with a minus sign, which is the cofactor sign of position $(1,2)$ .

Proof steps

\det A=a_{11}(\cdots)+a_{12}(\cdots)+a_{13}(\cdots)

Each of the six terms contains exactly one entry of row 1; collect the terms by that entry.

a_{11}(a_{22}a_{33}-a_{23}a_{32})=a_{11}M_{11}

The terms with $a_{11}$ are $a_{11}a_{22}a_{33}$ and . What multiplies is the minor .

-a_{12}(a_{21}a_{33}-a_{23}a_{31})=-a_{12}M_{12}

The terms with $a_{12}$ are $a_{12}a_{23}a_{31}$ and , which is minus times the minor .

a_{13}(a_{21}a_{32}-a_{22}a_{31})=a_{13}M_{13}

The terms with $a_{13}$ are $a_{13}a_{21}a_{32}$ and : that is times .

\det A=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}

The signs $+,-,+$ are exactly $(-1)^{1+j}$ , so each group is an entry of row 1 times its cofactor.

Practice

Crossing Out

To find $M_{ij}$ , cross out row $i$ and column $j$ and take the determinant of what is left.

A=\begin{pmatrix}2&-1&3\\4&0&1\\-2&5&6\end{pmatrix},\quad M_{12}=\begin{vmatrix}4&1\\-2&6\end{vmatrix}=26

Try it

Select the entries of $A$ that remain when the minor $M_{23}$ is formed.

$A=\begin{pmatrix}1 & 4 & 0 \\ 3 & -2 & 5 \\ 6 & 1 & 7\end{pmatrix}$

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Try it

For $A=\begin{pmatrix}2&-1&3\\4&0&1\\-2&5&6\end{pmatrix}$ , what is the minor $M_{12}$ ?

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The Chessboard of Signs

The cofactor sign $(-1)^{i+j}$ is $+$ where $i+j$ is even and $-$ where it is odd. Starting from in the top left, the signs alternate along every row and every column.

\begin{pmatrix}+&-&+&-\\-&+&-&+\\+&-&+&-\\-&+&-&+\end{pmatrix}

Try it

Select every position of a $3\times3$ matrix whose cofactor sign is $-$ .

$A=\begin{pmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{pmatrix}$

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Try it

For $A=\begin{pmatrix}1&2&3\\4&5&6\\0&1&2\end{pmatrix}$ , what is the cofactor $C_{21}$ ?

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A minor of a matrix is itself a matrix.

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Try it

What is the cofactor $C_{12}$ of $\begin{pmatrix}5&7\\2&3\end{pmatrix}$ ?

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Try it

Compute $\det\begin{pmatrix}3&0&2\\1&4&-1\\2&1&5\end{pmatrix}$ by expanding along the first row.

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Try it

A student expands a $3\times3$ determinant as $a_{11}M_{11}+a_{12}M_{12}+a_{13}M_{13}$ . What is wrong?

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What You Learned

The minor $M_{ij}$ is the determinant left after deleting row $i$ and column $j$ .
The cofactor is $C_{i j} = (- 1)^{i + j} M_{i j}$ , with signs in a chessboard pattern starting with .

Final checkpoint

Try it

For $A=\begin{pmatrix}1&2&0\\3&-1&4\\5&2&2\end{pmatrix}$ , what is the minor $M_{31}$ ?

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Try it

In a $4\times4$ matrix, what is the sign of the cofactor $C_{24}$ relative to the minor $M_{24}$ ?

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Try it

Compute $\det\begin{pmatrix}2&1&0\\1&3&1\\0&1&2\end{pmatrix}$ by expanding along the first row.

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Completion

Lesson complete

Great work! You now know how to:

form a minor by deleting a row and a column;
attach the cofactor sign from the chessboard pattern;
expand a $3\times3$ determinant along its first row;
explain where the expansion comes from.

Mapping the lesson.

Intuition