Matrices and their types

The essentials

A matrix is a rectangular arrangement of numbers in rows and columns. A matrix with m rows and n columns has order m × n and contains mn elements. The element in row i and column j is written aᵢⱼ. Two matrices are equal when they have the same order and every corresponding element is equal.

The transpose Aᵀ is obtained by turning rows into columns, so an m × n matrix has an n × m transpose. A square matrix A is symmetric if Aᵀ = A and skew-symmetric if Aᵀ = −A.

Key points

• Row matrix: one row. Column matrix: one column.
• Square matrix: number of rows equals number of columns.
• Diagonal matrix: square, with all non-diagonal elements zero.
• Scalar matrix: diagonal with all diagonal elements equal. Identity matrix I: scalar matrix with diagonal elements 1.
• Zero (null) matrix O: every element is 0.
• In a skew-symmetric matrix aᵢᵢ = −aᵢᵢ, so every diagonal element is 0.
• Every square matrix A can be written uniquely as ½(A + Aᵀ) + ½(A − Aᵀ), the sum of a symmetric and a skew-symmetric matrix.
• (Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ, (kA)ᵀ = kAᵀ.

Exam tips

• If a matrix has a prime number p of elements, its only possible orders are 1 × p and p × 1.
• When building a matrix from a rule for aᵢⱼ, write i for the row and j for the column carefully.

Common mistakes

• Writing the order as "columns × rows". Rows always come first.
• Assuming a diagonal matrix must have non-zero diagonal entries; zeros on the diagonal are allowed.