78. Minor

A minor of a matrix $𝐴$ is the determinant of a smaller square matrix obtained by deleting some rows and some columns of $𝐴$. The deleted rows and columns do not need to share indices — that distinguishes a minor from a principal minor.

Pick rows $\{1,3\}$ and columns $\{2,4\}$ — different sets, this is not a principal minor:

78.1. $\left(𝑖,𝑗\right)$-minor (cofactor expansion)

The most-used minor is the $\left(𝑖,𝑗\right)$-minor ${𝑀}_{𝑖𝑗}$: delete row $𝑖$ and column $𝑗$ from $𝐴$ and take the determinant of what’s left. Used in cofactor expansion:

78.2. Variants

• Principal minor — same row and column indices deleted
• Leading principal minor — keep only the top-left $𝑘\times 𝑘$ block
• Cofactor ${𝐶}_{𝑖𝑗}={\left(-1\right)}^{𝑖+𝑗}{𝑀}_{𝑖𝑗}$ — signed minor used in adjugate and inverse formulas

78.3. Where minors show up

• Determinant via cofactor expansion
• Adjugate matrix and the inverse formula ${𝐴}^{-1}=\frac{\text{adj}\left(𝐴\right)}{det\left(𝐴\right)}$
• Rank — equals the size of the largest non-zero minor
• Cramer’s Rule — see Cramer’s Rule