What is a matrix?
A matrix is a rectangular arrangement of numbers in rows and columns, written inside brackets. A matrix with m rows and n columns has order m × n and contains mn elements.
The element in the i-th row and j-th column is written aᵢⱼ. So a matrix can be written compactly as A = [aᵢⱼ]ₘₓₙ.
Matrices are used to organise data, to represent and solve systems of linear equations, and in computer graphics, cryptography and economics.
Types of matrices
Type | Description |
|---|---|
Row matrix | only one row (1 × n) |
Column matrix | only one column (m × 1) |
Square matrix | m = n |
Diagonal matrix | square, with every element off the main diagonal equal to 0 |
Scalar matrix | diagonal, with all diagonal elements equal |
Identity matrix I | diagonal, with all diagonal elements equal to 1 |
Zero matrix O | every element is 0 |
Two matrices are equal if they have the same order and every corresponding element is equal.
Operations on matrices
Addition and subtraction
Matrices of the same order are added (or subtracted) element by element: (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ.
Matrix addition is commutative (A + B = B + A) and associative. The zero matrix is the additive identity.
Scalar multiplication
Multiply every element by the scalar: (kA)ᵢⱼ = k aᵢⱼ.
Matrix multiplication
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
(AB)ᵢₖ = aᵢ₁b₁ₖ + aᵢ₂b₂ₖ + … + aᵢₙbₙₖ
That is, take the i-th row of A and the k-th column of B, multiply matching entries, and add.
Example. Let A = [1 2; 3 4] and B = [0 1; 1 0] (rows separated by semicolons).
AB = [1·0 + 2·1, 1·1 + 2·0; 3·0 + 4·1, 3·1 + 4·0] = [2 1; 4 3]
BA = [0·1 + 1·3, 0·2 + 1·4; 1·1 + 0·3, 1·2 + 0·4] = [3 4; 1 2]
So AB ≠ BA. Matrix multiplication is not commutative.
Other important facts:
Multiplication is associative, (AB)C = A(BC), and distributes over addition, A(B + C) = AB + AC.
AI = IA = A for a square matrix A of the same order as I.
AB = O does not mean A = O or B = O. Two non-zero matrices can multiply to give the zero matrix.
Transpose
The transpose Aᵀ (or A′) is obtained by turning rows into columns: (Aᵀ)ᵢⱼ = aⱼᵢ. An m × n matrix has an n × m transpose.
Properties:
(Aᵀ)ᵀ = A
(A + B)ᵀ = Aᵀ + Bᵀ
(kA)ᵀ = kAᵀ
(AB)ᵀ = BᵀAᵀ. The order reverses.
Symmetric and skew-symmetric matrices
A square matrix is symmetric if Aᵀ = A, that is aᵢⱼ = aⱼᵢ.
It is skew-symmetric if Aᵀ = −A, that is aᵢⱼ = −aⱼᵢ. Setting i = j gives aᵢᵢ = −aᵢᵢ, so every diagonal element of a skew-symmetric matrix is 0.
Every square matrix can be written uniquely as the sum of a symmetric and a skew-symmetric matrix:
A = ½(A + Aᵀ) + ½(A − Aᵀ)
The first part is symmetric and the second is skew-symmetric.
Invertible matrices
A square matrix A is invertible if there is a matrix B of the same order with AB = BA = I. Then B is the inverse of A, written A⁻¹.
The inverse, if it exists, is unique.
(AB)⁻¹ = B⁻¹A⁻¹, with the order reversed again.
A matrix is invertible exactly when its determinant is non-zero. The next chapter, on determinants, shows how to find A⁻¹.