78. Minor

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

Pick rows and columns — different sets, this is not a principal minor:

78.1. -minor (cofactor expansion)

The most-used minor is the -minor : delete row and column from 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 block
• Cofactor — signed minor used in adjugate and inverse formulas

78.3. Where minors show up

• Determinant via cofactor expansion
• Adjugate matrix and the inverse formula
• Rank — equals the size of the largest non-zero minor
• Cramer’s Rule — see Cramer’s Rule