Properties of Matrix Operations

Let matrices $A$ , $B$ , $C$ be conformable in size (wherever an operation requires it), and let $λ$ , $μ$ be arbitrary numbers (scalars; in general they may be complex). Below are the basic properties of addition, scalar multiplication, transposition, and multiplication of matrices.

Addition of Matrices

For matrices of the same size:

Commutativity: $A + B = B + A$
Associativity: $A + (B + C) = (A + B) + C$
Zero matrix: $A + O = A$ , where $O$ is the zero matrix of the same size as $A$ .
Opposite matrix: $A + (- A) = O$

Scalar Multiplication

Unit factor: $1 \cdot A = A$
Associativity: $λ (μ A) = (λ μ) A$
Distributivity over scalars: $(λ + μ) A = λ A + μ A$
Distributivity over matrices: $λ (A + B) = λ A + λ B$

Properties 1–8 mean that the set of matrices of a fixed size $m \times n$ forms a linear (vector) space.

Transposition

Double transposition: $(A^{T})^{T} = A$
Transpose of a sum: $(A + B)^{T} = A^{T} + B^{T}$
Pulling out a factor: $(λ A)^{T} = λ A^{T}$
Transpose of a product (the order reverses!): $(A \cdot B)^{T} = B^{T} \cdot A^{T}$

Matrix Multiplication

Associativity: $A \cdot (B \cdot C) = (A \cdot B) \cdot C$
Left distributivity: $A (B + C) = A B + A C$
Right distributivity: $(A + B) C = A C + B C$
Pulling out a scalar: $λ (A B) = (λ A) B = A (λ B)$ (when the product $A B$ is defined)
The identity matrix is a neutral element: $A E = A$ , $E A = A$ . In particular, for a square matrix $A E = E A = A$ .

Remark (non-commutativity). In general $A \cdot B \neq = B \cdot A$ . If $A B = B A$ , the matrices $A$ and $B$ are called commuting (or permutable). For example, any square matrix commutes with the identity and with itself.

Proof of Associativity of Multiplication (Property 13)

Let $A = (a_{ij})_{m \times n}$ , $B = (b_{j k})_{n \times p}$ , $C = (c_{k l})_{p \times q}$ . Denote $F = (A B) C$ and $G = A (B C)$ .

1. The sizes match.

(A_{m \times n} B_{n \times p}) C_{p \times q} = (A B)_{m \times p} C_{p \times q} = F_{m \times q},

A_{m \times n} (B_{n \times p} C_{p \times q}) = A_{m \times n} (B C)_{n \times q} = G_{m \times q} .

Both matrices have size $m \times q$ .

2. The corresponding entries match. By the definition of the product,

F_{i l} = k = 1 \sum p (A B)_{ik} c_{k l} = k = 1 \sum p (j = 1 \sum n a_{ij} b_{j k}) c_{k l} = k = 1 \sum p j = 1 \sum n a_{ij} b_{j k} c_{k l} .

G_{i l} = j = 1 \sum n a_{ij} (B C)_{j l} = j = 1 \sum n a_{ij} (k = 1 \sum p b_{j k} c_{k l}) = j = 1 \sum n k = 1 \sum p a_{ij} b_{j k} c_{k l} .

This is the same double sum (the order of summation may be swapped), so $F_{i l} = G_{i l}$ for all $i, l$ . Hence $(A B) C = A (B C)$ . $■$

On the Relation Between the Determinant and the Operations

The determinant is consistent with multiplication and transposition (for proofs, see the article Properties of Determinants):

det (A \cdot B) = det A \cdot det B, det (A^{T}) = det A .

At the same time, the determinant is in general not consistent with the sum: $det (A + B) \neq = det A + det B$ .