Multiplication of a matrix by a scalar

This lecture explains how to multiply a matrix by a scalar.

Table of contents

Definition
Properties
Solved exercises

Definition

Remember that a scalar is just a single number, that is, a matrix having dimension $1 \times 1$ .

Definition Let $A$ be a $K \times L$ matrix and $α$ be a scalar. The product of $A$ by $α$ is another $K \times L$ matrix, denoted by $α A$ , such that its $(k, l)$ -th entry is equal to the product of $α$ by the $(k, l)$ -th entry of $A$ , that is

(α A)_{k l} = α A_{k l}

for $1 \leq k \leq K$ and $1 \leq l \leq L$ .

The product $A α$ could be defined in the same manner. However, the order of the product does not really matter, because $a A_{k l} = A_{k l} a$ . Therefore, $A α$ can be considered the same as $α A$ .

Example Let $α = 2$ and define the $2 \times 3$ matrix

A = 012130

The product $α A$ is

α A = 2 \cdot 012130 = 2 \cdot 0 2 \cdot 1 2 \cdot 2 2 \cdot 1 2 \cdot 3 2 \cdot 0 = 024260

Properties

Basically all the properties enjoyed by multiplication of real numbers are inherited by multiplication of a matrix by a scalar.

Proposition (associative property) Multiplication of a matrix by a scalar is associative, that is,

α (β A) = (α β) A

for any matrix $A$ and any scalars $α$ and $β$ .

Proof

Let $A$ be a $K \times L$ matrix. We know that $β A$ is another $K \times L$ matrix, such that its $(k, l)$ -th entry is equal to the product of $β$ by the $(k, l)$ -th entry of $A$ , that is,

(β A)_{k l} = β A_{k l}

Furthermore, $α (β A)$ is a $K \times L$ matrix, such that its $(k, l)$ -th entry is equal to the product of $α$ by the $(k, l)$ -th entry of $β A$ , that is,

[α (β A)]_{k l} = α (β A)_{k l}

As a consequence, we have that

^{A}^{B}^{^{C}}^{^{D}} [α (β A)]_{k l} = α (β A)_{k l} = α β A_{k l} = (α β) A_{k l} = [(α β) A]_{k l}

where we have used the definitions of $α (β A)$ in step $A$ , $β A$ in step $B$ and $(α β) A$ in step $D$ . In step $C$ we have used the associativity of ordinary multiplication. Thus, we have proved that the $(k, l)$ -th entry of $α (β A)$ is equal to the $(k, l)$ -th entry of $(α β) A$ . Because this is true for every $k$ and $l$ , the statement is proved.

Proposition (distributive property 1) Multiplication of a matrix by a scalar is distributive with respect to matrix addition, that is,

α (A + B) = α A + α B

for any scalar $α$ and any matrices $A$ and $B$ such that their addition is meaningfully defined.

Proof

Let $A$ and $B$ be $K \times L$ matrices. By the definition of matrix addition $A + B$ is another $K \times L$ matrix, such that its $(k, l)$ -th entry is equal to the sum of the $(k, l)$ -th entry of $A$ and the $(k, l)$ -th entry of $B$ , that is,

(A + B)_{k l} = A_{k l} + B_{k l}

Furthermore, $α (A + B)$ is a $K \times L$ matrix, such that its $(k, l)$ -th entry is equal to the product of $α$ by the $(k, l)$ -th entry of $A + B$ , that is,

[α (A + B)]_{k l} = α (A + B)_{k l}

As a consequence, we have that

^{A}^{B}^{C}^{D}^{E} [α (A + B)]_{k l} = α (A + B)_{k l} = α (A_{k l} + B_{k l}) = α A_{k l} + α B_{k l} = (α A)_{k l} + (α B)_{k l} = (α A + α B)_{k l}

where we have used the definitions of $α (A + B)$ in step $A$ , $A + B$ in step $B$ , $α A$ and $α B$ in step $D$ and $α A + α B$ in step $E$ . In step $C$ we have used the distributivity of ordinary multiplication. Thus, we have proved that the $(k, l)$ -th entry of $α (A + B)$ is equal to the $(k, l)$ -th entry of $α A + α B$ . Because this is true for every $k$ and $l$ , the statement is proved.

Proposition (distributive property 2) Multiplication of a matrix by a scalar is distributive with respect to the addition of scalars, that is,

(α + β) A = α A + β A

for any scalars $α$ and $β$ and any matrix $A$ .

Proof

Let $A$ be a $K \times L$ matrix. We know that $(α + β) A$ is another $K \times L$ matrix, such that its $(k, l)$ -th entry is equal to the product of $α + β$ by the $(k, l)$ -th entry of $A$ , that is,

[(α + β) A]_{k l} = (α + β) A_{k l}

As a consequence, we have that

^{A}^{B}^{C}^{D} [(α + β) A]_{k l} = (α + β) A_{k l} = α A_{k l} + β A_{k l} = (α A)_{k l} + (β A)_{k l} = (α A + β A)_{k l}

where we have used the definitions of $(α + β) A$ in step $A$ , $α A$ and $β A$ in step $C$ , $α A + β A$ in step $D$ . In step $B$ we have used the distributivity of ordinary multiplication. Thus, we have proved that the $(k, l)$ -th entry of $(α + β) A$ is equal to the $(k, l)$ -th entry of $α A + β A$ . Because this is true for every $k$ and $l$ , the statement is proved.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let $A$ be the following $3 \times 3$ matrix

A = 210200003

Let $α = 3$ . Compute the product $α A$ .

Solution

The product $α A$ is another $3 \times 3$ matrix such that for each $1 \leq k \leq 3$ and $1 \leq l \leq 3$ , the $(k, l)$ -th element of $α A$ is equal to the product between $α$ and the $(k, l)$ -th element of $A$ :

α A = 3 \cdot 210200003 = 3 \cdot 2 3 \cdot 1 3 \cdot 0 3 \cdot 2 3 \cdot 0 3 \cdot 0 3 \cdot 0 3 \cdot 0 3 \cdot 3 = 630600009

Exercise 2

Let $A$ be a $1 \times 2$ row vector defined by

A = [13]

and $B$ a $2 \times 2$ matrix defined by

B = 2011

Compute the product

(A A^{⊤}) B

where $A^{⊤}$ denotes the transpose of $A$ .

Solution

The transpose of $A$ is

A^{⊤} = 13

The product between $A$ and its transpose is

A A^{⊤} = [13] 13 = 1 \cdot 1 + 3 \cdot 3 = 10

which is a scalar. As a consequence, we have that

(A A^{⊤}) B = 10 \cdot B = 10 \cdot 2011 = 10 \cdot 2 0 10 \cdot 1 10 \cdot 1 = 2001010

Exercise 3

Define two $1 \times 2$ row vectors:

A B = [12] = [01]

Find a scalar $α$ such that

A + α B = C

where

C = [10]

Solution

By applying the definition of multiplication of a matrix by a scalar, we obtain

α B = [0 α]

By applying the definition of matrix addition, we get

A + α B = [12] + [0 α] = [1 + 0 2 + α] = [1 2 + α]

Therefore, the equation

A + α B = C

is satisfied if and only if

[1 2 + α] = [10]

which in turn is satisfied if and only if

2 + α = 0

But this implies

α = - 2

How to cite

Please cite as:

Taboga, Marco (2021). "Multiplication of a matrix by a scalar", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/multiplication-of-a-matrix-by-a-scalar.