Algebra of matrices

The essentials

Matrices of the same order are added by adding corresponding elements, and a matrix is multiplied by a scalar k by multiplying every element by k. Addition is commutative and associative, the zero matrix is the additive identity, and −A is the additive inverse.

The product AB is defined only when the number of columns of A equals the number of rows of B. If A is m × n and B is n × p, then AB is m × p, and the element in row i, column j of AB is the sum of products of row i of A with column j of B.

Key points

• Matrix multiplication is associative, (AB)C = A(BC), and distributive, A(B + C) = AB + AC.
• It is not commutative: AB and BA may differ, or one of them may not even exist.
• AB = O is possible even when neither A nor B is the zero matrix.
• AI = IA = A for a square matrix A and identity I of the same order.
• (AB)ᵀ = BᵀAᵀ, with the order reversed.
• (A + B)² = A² + AB + BA + B², which equals A² + 2AB + B² only when AB = BA.

Exam tips

• Before multiplying, write the orders side by side; the inner numbers must match, and the outer numbers give the order of the product.
• For powers of a special matrix, compute A² and A³ and look for a pattern.

Common mistakes

• Cancelling matrices: AB = AC does not imply B = C in general.
• Using ordinary algebra identities such as (A + B)(A − B) = A² − B² without checking that A and B commute.