Matrix Scalar Multiplication

Understanding Matrix Scalar Multiplication

In the world of matrices, we not only deal with operations between matrices but also operations between a matrix and a single number. This single number is commonly referred to as a scalar.

Matrix scalar multiplication is one of the fundamental operations that is important to understand. Imagine you have a cake recipe, and you want to make twice as much. You would naturally multiply each ingredient's measurement by the number 2, right?

A similar concept applies to matrix scalar multiplication.

What is Matrix Scalar Multiplication?

Matrix scalar multiplication is the operation of multiplying every element in a matrix by a scalar number.

If we have a matrix $A$ and a scalar $k$ , then the result of the scalar multiplication of $k$ by matrix $A$ (written as $kA$ ) is a new matrix where each element is the product of the corresponding element of matrix $A$ and the scalar $k$ .

Mathematically, if matrix $A$ has an order of $m \times n$ :

A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}

Then the multiplication of matrix $A$ by scalar $k$ is:

kA = k \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} = \begin{bmatrix} k \cdot a_{11} & k \cdot a_{12} & \cdots & k \cdot a_{1n} \\ k \cdot a_{21} & k \cdot a_{22} & \cdots & k \cdot a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ k \cdot a_{m1} & k \cdot a_{m2} & \cdots & k \cdot a_{mn} \end{bmatrix}

The resulting matrix, $kA$ , will have the same order as matrix $A$ .

This concept is similar to repeated addition. For example, $2A$ is the same as $A + A$ . If we add matrix $A$ $k$ times, the result is $kA$ .

\underbrace{A + A + \cdots + A}_{k \text{ times}} = kA

Matrix Scalar Multiplication Examples

To better understand this concept, let's look at some examples.

Example 1:

Suppose we have matrix $P$ as in the reference image:

P = \begin{bmatrix} 9 & 7 \\ 3 & 1 \end{bmatrix}

Determine $2P$ !

Solution:

To calculate $2P$ , we multiply each element of matrix $P$ by the scalar 2.

2P = 2 \begin{bmatrix} 9 & 7 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} 2 \times 9 & 2 \times 7 \\ 2 \times 3 & 2 \times 1 \end{bmatrix} = \begin{bmatrix} 18 & 14 \\ 6 & 2 \end{bmatrix}

So, the result of $2P$ is $\begin{bmatrix} 18 & 14 \\ 6 & 2 \end{bmatrix}$ .

Example 2:

Given matrix $Q = \begin{bmatrix} -1 & \frac{1}{2} & 1 \\ 4 & -\frac{1}{4} & 2 \\ -6 & 1 & -4 \end{bmatrix}$ and scalar $k=4$ . Determine $4Q$ !

Solution:

We will multiply each element in matrix $Q$ by the scalar 4.

4Q = 4 \begin{bmatrix} -1 & \frac{1}{2} & 1 \\ 4 & -\frac{1}{4} & 2 \\ -6 & 1 & -4 \end{bmatrix}

= \begin{bmatrix} 4 \times (-1) & 4 \times \frac{1}{2} & 4 \times 1 \\ 4 \times 4 & 4 \times (-\frac{1}{4}) & 4 \times 2 \\ 4 \times (-6) & 4 \times 1 & 4 \times (-4) \end{bmatrix}

= \begin{bmatrix} -4 & 2 & 4 \\ 16 & -1 & 8 \\ -24 & 4 & -16 \end{bmatrix}

Thus, $4Q = \begin{bmatrix} -4 & 2 & 4 \\ 16 & -1 & 8 \\ -24 & 4 & -16 \end{bmatrix}$ .

Properties of Matrix Scalar Multiplication

Matrix scalar multiplication has several important properties to be aware of. Let $A$ and $B$ be matrices of the same order, $h$ and $k$ be scalars, and $O$ be the zero matrix.

Distributive over Matrix Addition:

$k(A + B) = kA + kB$

This means multiplying a scalar by the sum of two matrices is the same as summing the products of the scalar with each matrix.
Distributive over Scalar Addition:

$(h + k)A = hA + kA$

This means multiplying the sum of two scalars by a matrix is the same as summing the products of each scalar with the matrix.
Associative with Scalar Multiplication:

$(hk)A = h(kA) = k(hA)$

This means multiplying a matrix by the product of two scalars is the same as multiplying the first scalar by the product of the second scalar and the matrix.
Scalar Multiplication Identity:

$1A = A$

Multiplying a matrix by the scalar 1 does not change the matrix.
Multiplication by Zero Scalar:

$0A = O$

Multiplying a matrix by the scalar 0 results in the zero matrix ( $O$ ), which is a matrix where all elements are 0.
Multiplication of Zero Matrix by a Scalar:

$kO = O$

Multiplying the zero matrix by any scalar results in the zero matrix.
Multiplication by Scalar -1:

$(-1)A = -A$

Multiplying a matrix by the scalar -1 results in the negative of the matrix.

These properties help simplify calculations and provide a deeper understanding of matrix algebra.

Exercises

Given matrix $X = \begin{bmatrix} 5 & -2 \\ 0 & 4 \\ -1 & 7 \end{bmatrix}$ . Calculate $3X$ !
If $Y = \begin{bmatrix} 10 & 20 \\ -30 & 0 \end{bmatrix}$ , determine $\frac{1}{10}Y$ !
Given matrices $A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 5 \\ 6 & -2 \end{bmatrix}$ . Show that $3(A+B) = 3A + 3B$ !

Answer Key