Matrices
1.4 Determinants
Determinant of 2×2 and 3×3 matrices, minors, cofactors, Laplace expansion, and key properties including det(AB) = det(A)det(B).
1.4 Determinants
Every square matrix has an associated scalar called its determinant, written det(A) or |A|.
Definition 1.17 (1×1): det([a₁₁]) = a₁₁
Definition 1.18 (2×2): det(A) = a₁₁a₂₂ − a₁₂a₂₁
| 2 | −9 |
| 5 | 8 |
Minors and Cofactors
Definition 1.19 — Minor Mij: Determinant obtained by deleting row i and column j.
Definition 1.20 — Cofactor Cij: Cij = (−1)i+j Mij
Sign pattern of (−1)i+j:
+−+
−+−
+−+
For A = [[1,2,3],[7,5,6],[4,8,9]]:
M₁₁ = |5 6; 8 9| = 45−48 = −3 → C₁₁ = +M₁₁ = −3
M₁₂ = |7 6; 4 9| = 63−24 = 39 → C₁₂ = −M₁₂ = −39
M₃₂ = |1 3; 7 6| = 6−21 = −15 → C₃₂ = −M₃₂ = 15
Laplace Expansion (3×3)
det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ (expand along any row or column)
Example 1.12:
|4, 3, −1; 4, −7, 6; 1, 3, −9|
= 4·|−7,6;3,−9| − 3·|4,6;1,−9| + (−1)·|4,−7;1,3|
= 4(63−18) − 3(−36−6) − 1(12+7)
= 4(45) − 3(−42) − 19 = 180 + 126 − 19 = 287
Tip: When a row/column has zeros, expand along it to reduce computation. Example 1.14 uses column 3 (two zeros) for a shortcut.
Properties of Determinants
- det(AB) = det(A) · det(B)
- det(AT) = det(A)
- det(I) = 1
- det(λA) = λⁿ det(A) for n×n matrix A
- Swapping two rows/columns: det(B) = −det(A)
- Two identical rows/columns → det(A) = 0
Example 1.13: A = [[2,−7,3],[1,5,−1],[4,0,5]] → det(A) = 53, det(AT) = 53 ✓