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## Precalculus

### Unit 7: Lesson 3

Multiplying matrices by scalars

# Multiplying matrices by scalars

Learn how to find the result of a matrix multiplied by a real number.

## What you should be familiar with before taking this lesson

$\begin{array}{c} \goldE{\text{3 columns}} \\\\ \begin{array}{c} \blueE{\text{2 rows}}&\goldE{\LARGE\downarrow}&\goldE{\LARGE\downarrow}&\goldE{\LARGE\downarrow} \\\\ \begin{array}{c} \blueE{\LARGE\rightarrow} \\\\ \blueE{\LARGE\rightarrow}\end{array} &\left[\begin{array}{c} -2 \\\\ 5\end{array}\right. &\begin{array}{c}5 \\\\ 2\end{array} &\left.\begin{array}{c}6 \\\\ 7\end{array}\right] \end{array} \end{array}$
A matrix is a rectangular arrangement of numbers into rows and columns. Each number in a matrix is referred to as a matrix element or entry.
If this is new to you, you might want to check out our intro to matrices. You should also make sure you know how to add and subtract matrices.

## What you will learn in this lesson

We can multiply matrices by real numbers. This article explores how this works.

## Scalars and scalar multiplication

When we work with matrices, we refer to real numbers as scalars.
The term scalar multiplication refers to the product of a real number and a matrix. In scalar multiplication, each entry in the matrix is multiplied by the given scalar.
For example, given that $\bold A=\left[\begin{array}{c} 10 &6 \\\\ 4& 3 \end{array}\right]$, let's find 2, A.
To find 2, A, simply multiply each matrix entry by 2:
\begin{aligned} \greenD 2\bold A&=\greenD{2}\cdot{\left[\begin{array}{c} 10 &6 \\\\ 4& 3 \end{array}\right]} \\\\ &={\left[\begin{array}{c} \greenD2 \cdot10 &\greenD2\cdot 6 \\\\ \greenD2\cdot 4& \greenD2\cdot3 \end{array}\right]} \\\\ &=\left[\begin{array}{c} 20 &12 \\\\ 8& 6 \end{array}\right] \end{aligned}

Problem 1
Given $\bold B=\left[\begin{array}{c} -4 &-2 \\\\ 7& 1 \end{array}\right]$, find minus, 3, B.
minus, 3, B, equals

Problem 2
Given $\bold C=\left[\begin{array}{c} -42 \\\\ 27 \\\\ -3 \end{array}\right]$, find start fraction, 1, divided by, 3, end fraction, C.
start fraction, 1, divided by, 3, end fraction, C, equals

## Want to join the conversation?

• What about scalar division, ie division of a matrix by a real number? Will the same rules apply as in scalar multiplication?
• Short answer - yes, Absolutely!

Longer answer - You can view scalar division as multiplying by the reciprocal [i.e dividing a number/matrix by a set number is the same as multiplying by 1/number]
For example: 15/3 = 15*1/3.
Hence if you want to divide a matrix by a scalar simply multiply the matrix by the reciprocal of your divider (or just divide, its the same thing)
• what would a matrice multiplied by a zero be?
• You would get a matrix where every entry is 0. This is called a zero matrix.
• In the definition of scalar multiplication the scalar must be a real number. But what happens when that scalar is an imaginary number? Will the multiplication of the imaginary number with the matrix be defined as undefined?
• I was solving a Khan Academy problem where I had to solve for X. There were two matrices of equal dimensions (3x2) and X is in one of them. I had to add the matrices together and multiply by 2. How come it displays the answer as a 3x3 dimensional matrix when the answer is actually a 3x2 matrix? I looked at the help section and it stated that the answer is actually a 3x2 answer.
• The Khan Academy problem section simply allocates more space than needed in thier problems, simply fill out the parts of the matrix you need to fill out and leave other cells blank.
• what are scalar numbers?
• Scalar are just your basic numbers. 1, 2, 5, 1000, π are all scalars.
• can anyone explain clearly with example on how to do fractional scalar division?
• With a fractional scalar division you would kind of divide the numbers for example - let the scalar be 1/3 and the number in the matrix be 12 you would just put a 1 under the 12 to make it a fraction, then just multiply, giving you 4.
• So scalar division exists too, and we can use it, but we more often just multiply by the reciprocal, correct?
• That's kind of the definition of division, multiplying by the reciprocal. Or at least a definition. But yeah, you get it.