Matrices
Matrix questions test order and equality, the three elementary operations, transpose, symmetric and skew-symmetric matrices, and the adjoint–inverse method. Expect a definition-based one-mark item and a computation of A⁻¹ or a verification of (AB)′ = B′A′ for 3–4 marks.
Rules to know
- Two matrices are equal only if they have the same order and matching entries.
- Addition needs identical orders; multiplication of A (m×n) by B (p×q) needs n = p, giving order m×q.
- (A′)′ = A, (A + B)′ = A′ + B′, (AB)′ = B′A′.
- A is symmetric if A′ = A and skew-symmetric if A′ = −A; every square matrix splits as A = ½(A + A′) + ½(A − A′).
- A⁻¹ = adj(A)/|A|, valid only when |A| ≠ 0.
Worked example
For A = [[2, 3], [1, 2]], |A| = (2)(2) − (3)(1) = 1 ≠ 0, so A is invertible. For a 2×2 matrix, adj(A) = [[d, −b], [−c, a]] = [[2, −3], [−1, 2]]. Therefore A⁻¹ = adj(A)/|A| = [[2, −3], [−1, 2]]. Check: multiplying A by this gives the identity, confirming the result.
Traps
- Writing (AB)′ = A′B′ instead of B′A′.
- Assuming AB = BA; matrix multiplication is not commutative.
- Dividing by |A| without first checking it is non-zero.
- Mixing up the sign pattern of the adjoint of a 2×2 matrix.