Cofactor Expansion¶

Theorem¶

For a square matrix $\boldsymbol{A}$ , let $\boldsymbol{A}_{ij}$ be the matrix obtained by deleting row $i$ and column $j$ . The number $\det(\boldsymbol{A}_{ij})$ is the minor of entry $(i,j)$ , and

C_{ij}=(-1)^{i+j}\det(\boldsymbol{A}_{ij})

is the corresponding cofactor.

The determinant can be computed by expanding along any row or column:

\det(\boldsymbol{A})=\sum_k a_{ik}C_{ik}

for expansion along row $i$ , or

\det(\boldsymbol{A})=\sum_k a_{kj}C_{kj}

for expansion along column $j$ .

Interpretation¶

Cofactor expansion reduces the determinant of an $n\times n$ matrix to $n$ determinants of size $(n-1)\times(n-1)$ $1)$ , each multiplied by an entry of the chosen row or column and by a sign. Repeating the step ends at $1\times1$ determinants; the determinant itself is defined this way, along the first row.

A 3 by 3 matrix with its first row highlighted for expansion and the minor of the entry a11 shown as the remaining block, next to the checkerboard pattern of signs plus and minus.

Expanding a $3\times3$ matrix along row $1$ gives $\det \boldsymbol{A}=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}$ $a_{13} C_{13}$ . Deleting row $1$ and column $1$ (shaded) leaves the white $2\times2$ block $\boldsymbol{A}_{11}$ , whose determinant is the minor of $a_{11}$ . The sign $(-1)^{i+j}$ of each cofactor is read off the checkerboard on the right: $+$ on the main diagonal, alternating along every row and column.

Since every row and every column gives the same value, the choice is purely practical: an entry $a_{ik}=0$ removes the term $a_{ik}C_{ik}$ without computing its minor. A full expansion of an $n\times n$ determinant produces $n!$ products, so for larger matrices one first creates zeros, for example by adding a multiple of one row to another, which does not change the determinant, and then expands along the sparse row or column.

For $3\times3$ matrices the expansion gives the same six products as Sarrus' rule, and the cofactors are the entries of the adjugate matrix, which gives a formula for the inverse.

A common slip is to drop the sign: $C_{12}=-\det(\boldsymbol{A}_{12})$ $- det (A_{12})$ . The sign depends only on the position $(i,j)$ , not on the sign of the entry $a_{ij}$ .

Key Results¶

Cofactor expansion can be performed along any row or column.
Choosing a row or column with many zeros makes the computation shorter.

Examples¶

For a $3\times3$ matrix, expanding along the first row gives three $2\times2$ determinants.
For a diagonal matrix, expanding along sparse rows quickly gives the product of diagonal entries.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 4

Aliases: Laplace expansion, expansion by minors Keywords: minor, cofactor, expansion along a row, expansion along a column Symbols: sum

Last updated: 20 October 2026 03:02