Cofactor Expansion and Cramer's Rule

Cofactor Expansion

Definition 75: Cofactor Matrix

Let $𝐴 \in 𝐹^{𝑛 \times 𝑛}$ be a square matrix. For any $𝑖, 𝑗 \in [𝑛]$ , the minor of $𝐴$ corresponding to the entry $𝑎_{𝑖 𝑗}$ , denoted $𝑚_{𝑖 𝑗}$ , is the determinant of the $(𝑛 - 1) \times (𝑛 - 1)$ matrix $𝐴_{𝑖 𝑗}$ obtained by deleting the $𝑖$ ^th row and $𝑗$ ^th column of $𝐴$ :

𝑚_{𝑖 𝑗} = \det (𝐴_{𝑖 𝑗}) .

The cofactor matrix $𝐶$ of $𝐴$ is the $𝑛 \times 𝑛$ matrix whose $(𝑖, 𝑗)$ ^th entry is the cofactor $𝑐_{𝑖 𝑗}$ , defined by:

𝑐_{𝑖 𝑗} = {(- 1)}^{𝑖 + 𝑗} 𝑚_{𝑖 𝑗} = {(- 1)}^{𝑖 + 𝑗} \det (𝐴_{𝑖 𝑗}) .

Theorem 66: Cofactor Expansion Theorem

Fix the determinant of a $1 \times 1$ matrix $\det ((\begin{matrix} 𝑎_{11} \end{matrix})) = 𝑎$ . Let $𝐴 \in 𝐹^{𝑛 \times 𝑛}$ be a square matrix for $𝑛 \geq 2$ , and let $𝐶 = (𝑐_{𝑖 𝑗})$ be its cofactor matrix. Then the determinant of $𝐴$ can be computed by either recursive formula:

Expansion Along the $𝒊$ ^th Row.
$\det (𝐴) = \sum_{𝑗 = 1}^{𝑛} 𝑎_{𝑖 𝑗} 𝑐_{𝑖 𝑗} = \sum_{𝑗 = 1}^{𝑛} 𝑎_{𝑖 𝑗} {(- 1)}^{𝑖 + 𝑗} \det (𝐴_{𝑖 𝑗}) for any fixed 𝑖 \in [𝑛] .$
Expansion Along the $𝒋$ ^th Column.
$\det (𝐴) = \sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖 𝑗} 𝑐_{𝑖 𝑗} = \sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖 𝑗} {(- 1)}^{𝑖 + 𝑗} \det (𝐴_{𝑖 𝑗}) for any fixed 𝑗 \in [𝑛] .$