Properties of matrices — A/L Combined Mathematics

A matrix is a rectangular array of numbers arranged in m rows and n columns, said to have order m × n. Two matrices A and B can be added or subtracted if and only if they have identical orders (entry-wise operation: [A ± B]_ij = a_ij ± b_ij). Scalar multiplication kA multiplies every element of A by scalar k. Matrix multiplication AB is defined if and only if the number of columns in A equals the number of rows in B: if A is m × p and B is p × n, then the product C = AB is m × n with elements c_ij = ∑_k=1^p a_ik b_kj. Matrix multiplication is associative [A(BC) = (AB)C] and distributive [A(B + C) = AB + AC], but non-commutative in general (AB ≠ BA).

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

1. Order Compatibility Check: Verifying dimension compatibility before attempting matrix multiplication.
2. Non-Commutativity Demonstration: Showing that AB ≠ BA by direct computation of both products for given 2 × 2 matrices.
3. Row-by-Column Multiplication Precision: Carefully calculating sum-of-products for each entry without sign errors.