Definition:Cofactor Matrix

Definition

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.

Let $A_{r s}$ denote the cofactor of the element whose indices are $\tuple {r, s}$.

The cofactor matrix of $\mathbf A$ is the square matrix of order $n$:

$\quad \mathbf C = \begin {bmatrix} A_{1 1} & A_{1 2} & \cdots & A_{1 n} \\ A_{2 1} & A_{2 2} & \cdots & A_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{n 1} & A_{n 2} & \cdots & A_{n n} \\ \end {bmatrix}$

Also known as

The cofactor matrix is also called comatrix or matrix of cofactors.

Also see

• Definition:Adjugate Matrix

• Matrix Product with Adjugate Matrix

• Results about cofactor matrices can be found here.

Sources

• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.7$ Minors and cofactors
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): matrix of cofactors