Matrix Determinant
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det(A)
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Invertible—
Sign—
|det(A)|—
Determinant Formulas & Properties
2×2 Formula
det = ad - bc
Product of main diagonal minus product of anti-diagonal.
3×3 Formula (Sarrus)
Expand along row 1: det = a(ei-fh) - b(di-fg) + c(dh-eg)
Transpose Invariance
det(A) = det(Aᵀ) — the determinant of the transpose equals the original.
Product Property
det(AB) = det(A) × det(B) — determinant of a product is the product of determinants.
Row/Column Operations
Swapping two rows/cols flips the sign. Multiplying a row by k multiplies det by k.
Invertibility
A matrix is invertible ⟺ det(A) ≠ 0. Zero determinant means the matrix is singular.