Special square matrices — A/L Combined Mathematics

A square matrix has an equal number of rows and columns (n × n). Key special square matrices include: (1) Identity Matrix I: Main diagonal entries are 1 and all off-diagonal entries are 0, satisfying AI = IA = A; (2) Zero Matrix O: All entries are 0, satisfying A + O = A and AO = OA = O; (3) Diagonal Matrix: All off-diagonal entries are zero; (4) Symmetric Matrix: Satisfies A^T = A, where a_ij = a_ji for all i, j; (5) Skew-Symmetric Matrix: Satisfies A^T = -A, requiring a_ij = -a_ji, which strictly mandates that all principal diagonal entries are zero (a_ii = 0). The determinant of a 2 × 2 matrix A = [[a, b], [c, d]] is defined as det(A) = |A| = ad - bc.

Check before you answer (A/L Pure Maths Paper 1)

1. Determinant Calculation: Accurately calculating det(A) = ad - bc and determining whether the matrix is singular (det(A) = 0) or non-singular.
2. Skew-Symmetric Diagonal Zero Condition: Proving that diagonal elements must be 0 from a_ii = -a_ii ⟹ 2a_ii = 0.
3. Matrix Polynomial Evaluation: Evaluating matrix polynomial expressions like A² - (a + d)A + (ad - bc)I = O (Cayley-Hamilton identity for 2 × 2 matrices).