Determinants

54

For an n×n matrix A, if we multiply by scalar k, then det(kA) = k^n × det(A). Here n=2, so det(3A) = 3² × 6 = 9 × 6 = 54. Common mistake: Calculating 3 × 6 = 18 instead of 3² × 6.

det(AB) = det(A) × det(B), det(A^T) = det(A), If two rows are identical, det(A) = 0

Correct properties: det(AB) = det(A)det(B), det(A^T) = det(A), and identical rows make det = 0. Wrong: det(A+B) ≠ det(A)+det(B) and det(kA) = k^n×det(A), not k×det(A).

-6

Minor M₁₂ is obtained by deleting row 1 and column 2, giving matrix [[4, 6], [7, 9]]. M₁₂ = det([[4, 6], [7, 9]]) = 4×9 - 6×7 = 36 - 42 = -6.

8

Cofactor A₁₁ = (-1)^(1+1) × M₁₁ = (-1)² × M₁₁ = 1 × M₁₁ = M₁₁. Since A₁₁ = 8, therefore M₁₁ = 8. The sign factor (-1)^(i+j) for position (1,1) is positive.