Matrices and Determinants
What is tested
Matrices and Determinants asks for order and types of matrices, transpose and symmetry, evaluation and properties of determinants, adjoint and inverse, and the solution of a linear system by the matrix or determinant method.
Core rules
- (AB)ᵀ = BᵀAᵀ; A is symmetric if Aᵀ = A and skew-symmetric if Aᵀ = −A, which forces every diagonal entry to be 0.
- Expansion along a row: |A| = Σ aᵢⱼCᵢⱼ with Cᵢⱼ = (−1)^(i+j) Mᵢⱼ.
- |AB| = |A||B|, |Aᵀ| = |A|, |kA| = kⁿ|A| for order n, |adj A| = |A|^(n−1).
- A⁻¹ = adj A/|A|, existing only when |A| ≠ 0; (AB)⁻¹ = B⁻¹A⁻¹.
- The system AX = B has unique solution X = A⁻¹B when |A| ≠ 0.
Worked example
A = [[2, 3],[1, 4]]. |A| = 2·4 − 3·1 = 5 ≠ 0, so A is invertible. For a 2×2 matrix, adj A = [[d, −b],[−c, a]] = [[4, −3],[−1, 2]]. Hence A⁻¹ = (1/5)[[4, −3],[−1, 2]]. Check: [[2,3],[1,4]] · (1/5)[[4,−3],[−1,2]] = (1/5)[[8−3, −6+6],[4−4, −3+8]] = [[1,0],[0,1]], confirming the inverse.
Marks-losing traps
Assuming AB = BA; expanding (A+B)² as A² + 2AB + B²; dropping the sign (−1)^(i+j) in a cofactor; forgetting that adj A is the transpose of the cofactor matrix; claiming an inverse when |A| = 0; swapping two rows of a determinant without flipping its sign.