Multiplication of a matrix by a scalar

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

Definition

Remember that a scalar is just a single number, that is, a matrix having dimension .

Definition Let be a matrix and be a scalar. The product of by is another matrix, denoted by , such that its -th entry is equal to the product of by the -th entry of , that isfor and .

The product could be defined in the same manner. However, the order of the product does not really matter, because . Therefore, can be considered the same as .

Example Let and define the matrixThe product is

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,for any matrix and any scalars and .

Proof

Let be a matrix. We know that is another matrix, such that its -th entry is equal to the product of by the -th entry of , that is,Furthermore, is a matrix, such that its -th entry is equal to the product of by the -th entry of , that is,As a consequence, we have thatThus, we have proved that the -th entry of is equal to the -th entry of . Because this is true for every and , the statement is proved.

Proposition (distributive property 1) Multiplication of a matrix by a scalar is distributive with respect to matrix addition, that is,for any scalar and any matrices and such that their addition is meaningfully defined.

Proof

Let and be matrices. By the definition of matrix addition is another matrix, such that its -th entry is equal to the sum of the -th entry of and the -th entry of , that is,Furthermore, is a matrix, such that its -th entry is equal to the product of by the -th entry of , that is,As a consequence, we have thatThus, we have proved that the -th entry of is equal to the -th entry of . Because this is true for every and , 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,for any scalars and and any matrix .

Proof

Let be a matrix. We know that is another matrix, such that its -th entry is equal to the product of by the -th entry of , that is,As a consequence, we have thatThus, we have proved that the -th entry of is equal to the -th entry of . Because this is true for every and , the statement is proved.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let be the following matrixLet . Compute the product .

Solution

The product is another matrix such that for each and , the -th element of is equal to the product between and the -th element of :

Exercise 2

Let be a row vector defined byand a matrix defined byCompute the productwhere denotes the transpose of .

Solution

The transpose of isThe product between and its transpose iswhich is a scalar. As a consequence, we have that

Exercise 3

Define two row vectors:Find a scalar such thatwhere

Solution

By applying the definition of multiplication of a matrix by a scalar, we obtainBy applying the definition of matrix addition, we getTherefore, the equationis satisfied if and only ifwhich in turn is satisfied if and only ifBut this implies

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