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Multiplication of a matrix by a scalar

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

Table of Contents

Definition

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

Definition Let A be a $K	imes L$ matrix and $lpha $ be a scalar. The product of A by $lpha $ is another $K	imes L$ matrix, denoted by $lpha A$, such that its $left( k,l
ight) $-th entry is equal to the product of $lpha $ by the $left( k,l
ight) $-th entry of A, that is[eq1]for $1leq kleq K$ and $1leq lleq L$.

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

Example Let $lpha =2$ and define the $2	imes 3$ matrix[eq2]The product $lpha A$ is[eq3]

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,[eq4]for any matrix A and any scalars $lpha $ and $eta $.

Proof

Let A be a $K	imes L$ matrix. We know that $eta A$ is another $K	imes L $ matrix, such that its $left( k,l
ight) $-th entry is equal to the product of $eta $ by the $left( k,l
ight) $-th entry of A, that is,[eq5]Furthermore, [eq6] is a $K	imes L$ matrix, such that its $left( k,l
ight) $-th entry is equal to the product of $lpha $ by the $left( k,l
ight) $-th entry of $eta A$, that is,[eq7]As a consequence, we have that[eq8]Thus, we have proved that the $left( k,l
ight) $-th entry of [eq9] is equal to the $left( k,l
ight) $-th entry of [eq10]. 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,[eq11]for any scalar $lpha $ and any matrices A and $B$ such that their addition is meaningfully defined.

Proof

Let A and $B$ be $K	imes L$ matrices. By the definition of matrix addition $A+B$ is another $K	imes L$ matrix, such that its $left( k,l
ight) $-th entry is equal to the sum of the $left( k,l
ight) $-th entry of A and the $left( k,l
ight) $-th entry of $B$, that is,[eq12]Furthermore, [eq13] is a $K	imes L$ matrix, such that its $left( k,l
ight) $-th entry is equal to the product of $lpha $ by the $left( k,l
ight) $-th entry of $A+B$, that is,[eq14]As a consequence, we have that[eq15]Thus, we have proved that the $left( k,l
ight) $-th entry of [eq16] is equal to the $left( k,l
ight) $-th entry of $lpha A+lpha B$. 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,[eq17]for any scalars $lpha $ and $eta $ and any matrix A.

Proof

Let A be a $K	imes L$ matrix. We know that [eq18] is another $K	imes L$ matrix, such that its $left( k,l
ight) $-th entry is equal to the product of $lpha +eta $ by the $left( k,l
ight) $-th entry of A, that is,[eq19]As a consequence, we have that[eq20]Thus, we have proved that the $left( k,l
ight) $-th entry of [eq21] is equal to the $left( k,l
ight) $-th entry of $lpha A+eta A$. 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 $3	imes 3$ matrix[eq22]Let $lpha =3$. Compute the product $lpha A$.

Solution

The product $lpha A$ is another $3	imes 3$ matrix such that for each $1leq kleq 3$ and $1leq lleq 3$, the $left( k,l
ight) $-th element of $lpha A$ is equal to the product between $lpha $ and the $left( k,l
ight) $-th element of A: [eq23]

Exercise 2

Let A be a $1	imes 2$ row vector defined by[eq24]and $B$ a $2	imes 2$ matrix defined by[eq25]Compute the product[eq26]where $A^{	op }$ denotes the transpose of A.

Solution

The transpose of A is[eq27]The product between A and its transpose is[eq28]which is a scalar. As a consequence, we have that[eq29]

Exercise 3

Define two $1	imes 2$ row vectors:[eq30]Find a scalar $lpha $ such that[eq31]where[eq32]

Solution

By applying the definition of multiplication of a matrix by a scalar, we obtain[eq33]By applying the definition of matrix addition, we get[eq34]Therefore, the equation[eq35]is satisfied if and only if[eq36]which in turn is satisfied if and only if[eq37]But this implies[eq38]

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