Matrices and matrix algebra

Maths · Matrices and determinants. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A matrix is a rectangular array of numbers. A matrix with m rows and n columns has order m × n (rows first). Matrices are named by their shape and entries: row, column, square, diagonal, scalar, identity, null, and so on.

Two matrices can be added or subtracted only when they have the same order. The product AB is defined only when the number of columns of A equals the number of rows of B. In general AB ≠ BA.

The transpose A^t is found by changing rows into columns. A square matrix has an inverse only when its determinant is not zero.

Rules to remember

If A is m × n and B is n × p, then AB is m × p. Each entry is (a row of A) × (a column of B), multiplied term by term and added.
Symmetric: A^t = A. Skew-symmetric: A^t = −A (every diagonal entry is 0).
Hermitian: (Ā)^t = A. Skew-Hermitian: (Ā)^t = −A. Here Ā is the matrix of conjugates.
Diagonal matrix: square, every entry off the main diagonal is 0. Scalar matrix: a diagonal matrix with all diagonal entries equal. Identity matrix I: diagonal entries all 1.
For the 2 × 2 matrix with rows [a b] and [c d]: |A| = ad − bc and adj A has rows [d −b] and [−c a].
A⁻¹ = (1/|A|) × adj A, provided |A| ≠ 0; then AA⁻¹ = A⁻¹A = I.
(AB)^t = B^tA^t and (AB)⁻¹ = B⁻¹A⁻¹ (the order reverses).

Common mistake

For the adjoint of a 2 × 2 matrix students change the wrong entries. Interchange the two main diagonal entries and change the signs of the other two; do not move the other two.

Worked example

Find the inverse of the matrix A with rows [4 3] and [2 2].

|A| = (4 × 2) − (3 × 2) = 8 − 6 = 2, which is not zero, so the inverse exists.
adj A has rows [2 −3] and [−2 4].
A⁻¹ = (1/2) × adj A, with rows [1 −3/2] and [−1 2].
Check the first row of AA⁻¹: 4 × 1 + 3 × (−1) = 1 and 4 × (−3/2) + 3 × 2 = 0.

Answer: A⁻¹ is the matrix with rows [1 −3/2] and [−1 2].

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