Definition · applies to square matrices · also called $C_{ij}$

High school

Consider a $3 \times 3$ matrix $A$ and a row number $i$ and a column number $j$. The $(i, j)$ cofactor of $A$ is the $(i, j)$ minor of $A$, with its sign changed when $i + j$ is odd, following the checkerboard pattern

\[\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}.\]
Undergraduate

Consider an $n \times n$ matrix $A$ and indices $i$ and $j$. The $(i, j)$ cofactor of $A$ is the $(i, j)$ minor $M_{ij}$ of $A$ multiplied by $(-1)^{i + j}$:

\[C_{ij} = (-1)^{i + j} M_{ij}.\]

Examples

Worked examples

  • The matrix \(A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{bmatrix}\) : $C_{11} = 2$, $C_{12} = 2$, $C_{13} = -3$ Undergraduate
    The cofactors for the first row of the matrix \[A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{bmatrix}\] are the minors $2$, $-2$, and $-3$ with the signs $+$, $-$, and $+$, so $C_{12} = -M_{12} = 2$, while $C_{11} = M_{11}$ and $C_{13} = M_{13}$.
  • The matrix \(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\) : $C_{11} = 3$, $C_{12} = -4$, $C_{21} = -1$, $C_{22} = 2$ Undergraduate
    The cofactors of \(A = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\) are its minors with the signs $+$ and $-$ in a checkerboard pattern, so $C_{12} = -M_{12} = -4$ and $C_{21} = -M_{21} = -1$. Expanding along row $1$ gives $\det A = 2 \cdot 3 + 1 \cdot (-4) = 2$.

Properties

  • The determinant of $A$ is the sum of the entries of one row times their cofactors (the Laplace, or cofactor, expansion): $\det A = \sum_{j = 1}^{n} a_{ij} C_{ij}$ for each row $i$, where $a_{ij}$ is the $(i, j)$ entry of $A$, and similarly for each column.
  • The transpose of the matrix of cofactors is the adjugate $\operatorname{adj} A$ of $A$, which satisfies $A \operatorname{adj} A = (\det A) I$. So if $\det A \neq 0$, then $A^{-1} = \operatorname{adj} A / \det A$.
  • The cofactor $C_{ij}$ is the partial derivative of $\det A$ with respect to the entry $a_{ij}$.

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