Cofactor Expansion Method

Reducing a Determinant to Smaller Determinants

Cofactor expansion breaks the determinant of a square matrix into smaller determinants. Choose one row or column, multiply each entry in that row or column by its corresponding cofactor, and add the products. This method can be used for matrices larger than $2 \times 2$ .

Minor of a Matrix Element

Each entry $a_{ij}$ in a square matrix has a minor $M_{ij}$ .

To calculate $M_{ij}$ , cross out row $i$ and column $j$ , then calculate the determinant of the remaining submatrix.

Finding the Minor

Suppose we have a $3 \times 3$ matrix $A$ as follows:

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

If we want to find the minor of the element $a_{22}$ (i.e., $M_{22}$ ), we remove the second row and second column from matrix $A$ :

Cross out the second row and the second column as follows:

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

The matrix remaining after the removal is:

\begin{bmatrix} a_{11} & a_{13} \\ a_{31} & a_{33} \end{bmatrix}

The minor $M_{22}$ is the determinant of this remaining matrix:

\begin{aligned} M_{22} &= \det \begin{pmatrix} a_{11} & a_{13} \\ a_{31} & a_{33} \end{pmatrix} \\ &= (a_{11} \times a_{33}) - (a_{13} \times a_{31}) \end{aligned}

Cofactor of a Matrix Element

The cofactor of entry $a_{ij}$ is written as $k_{ij}$ or $C_{ij}$ and calculated from the minor $M_{ij}$ :

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

The parity of $i+j$ determines the cofactor sign:

If the sum $i+j$ is an even number, then $(-1)^{i+j} = 1$ . This means $k_{ij} = M_{ij}$ (the cofactor is equal to its minor).
If the sum $i+j$ is an odd number, then $(-1)^{i+j} = -1$ . This means $k_{ij} = -M_{ij}$ (the cofactor is the negative of its minor).

For a $3 \times 3$ matrix, the sign pattern of $(-1)^{i+j}$ is:

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

This sign indicates whether the cofactor will be equal to its minor (sign $+$ ) or the negative of its minor (sign $-$ ).

Finding the Cofactor

The following calculation continues the example of matrix $A$ and minor $M_{22}$ .

The cofactor for element $a_{22}$ is $k_{22}$ .

Since $i=2$ and $j=2$ , then $i+j = 2+2 = 4$ (even).

k_{22} = (-1)^{2+2} M_{22} = (1) M_{22} = M_{22}

Next, we calculate cofactor $k_{12}$ for entry $a_{12}$ .

First, we find its minor, $M_{12}$ , by removing the first row and second column from matrix $A$ :

\begin{aligned} M_{12} &= \det \begin{pmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{pmatrix} \\ &= (a_{21} \times a_{33}) - (a_{23} \times a_{31}) \end{aligned}

Then, we calculate its cofactor. For $a_{12}$ , $i=1$ and $j=2$ , so $i+j = 1+2 = 3$ (odd).

\begin{aligned} k_{12} &= (-1)^{1+2} M_{12} \\ &= (-1) M_{12} \\ &= -M_{12} \end{aligned}

Calculating the Determinant using Cofactor Expansion

To calculate the determinant of matrix $A$ by cofactor expansion, choose one row or column. Multiply each selected entry by its corresponding cofactor, then add the products. The determinant of $A$ is written as $\det(A)$ or $|A|$ .

Formula for Cofactor Expansion Along the $i$ -th Row:

\begin{aligned} &\det(A) \\ &= |A| \\ &= \sum_{j=1}^{n} a_{ij}k_{ij} \\ &= a_{i1}k_{i1} + a_{i2}k_{i2} + \dots + a_{in}k_{in} \end{aligned}

This means we choose the $i$ -th row. Then, for each column $j$ in that row, we multiply the element $a_{ij}$ by its cofactor $k_{ij}$ , and sum them all.

Formula for Cofactor Expansion Along the $j$ -th Column:

\begin{aligned} &\det(A) \\ &= |A| \\ &= \sum_{i=1}^{n} a_{ij}k_{ij} \\ &= a_{1j}k_{1j} + a_{2j}k_{2j} + \dots + a_{nj}k_{nj} \end{aligned}

This means we choose the $j$ -th column. Then, for each row $i$ in that column, we multiply the element $a_{ij}$ by its cofactor $k_{ij}$ , and sum them all.

Expanding along any row or column gives the same determinant.

Choose a row or column with many zeros when one is available. Every zero eliminates its product, so fewer terms remain to calculate.

Example of Determinant Calculation

Calculate the determinant of the following matrix $P$ using the cofactor expansion method:

P = \begin{bmatrix} 1 & 3 & 2 \\ 2 & 6 & 2 \\ 5 & 9 & 4 \end{bmatrix}

We will perform cofactor expansion along the first row (i.e., $i=1$ ).

Based on the formula, the determinant of $P$ is:

\det(P) = a_{11}k_{11} + a_{12}k_{12} + a_{13}k_{13}

From matrix $P$ , the elements of the first row are:

$a_{11} = 1$
$a_{12} = 3$
$a_{13} = 2$

Now, we need to calculate the cofactors $k_{11}$ , $k_{12}$ , and $k_{13}$ .

Calculating $k_{11}$ ( $i=1, j=1$ , so $i+j=2$ , even):
$\begin{aligned} M_{11} &= \det \begin{pmatrix} 6 & 2 \\ 9 & 4 \end{pmatrix} \\ &= (6 \times 4) - (2 \times 9) \\ &= 24 - 18 \\ &= 6 \end{aligned}$
Calculating $k_{12}$ ( $i=1, j=2$ , so $i+j=3$ , odd):
$\begin{aligned} M_{12} &= \det \begin{pmatrix} 2 & 2 \\ 5 & 4 \end{pmatrix} \\ &= (2 \times 4) - (2 \times 5) \\ &= 8 - 10 \\ &= -2 \end{aligned}$
Calculating $k_{13}$ ( $i=1, j=3$ , so $i+j=4$ , even):
$\begin{aligned} M_{13} &= \det \begin{pmatrix} 2 & 6 \\ 5 & 9 \end{pmatrix} \\ &= (2 \times 9) - (6 \times 5) \\ &= 18 - 30 \\ &= -12 \end{aligned}$

After all cofactors are obtained, we substitute them back into the determinant formula:

\det(P) = a_{11}k_{11} + a_{12}k_{12} + a_{13}k_{13}

So, the determinant of matrix $P$ is $-12$ .