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)

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 would a matrice multiplied by a zero be?

You would get a matrix where every entry is 0. This is called a zero matrix.

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.

what are scalar numbers?

Scalar are just your basic numbers. 1, 2, 5, 1000, π are all scalars.

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.

Main content

Algebra (all content)

Course: Algebra (all content) > Unit 20

Lesson 6: Multiplying matrices by scalars

Multiplying matrices by scalars

Google Classroom

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}

Check your understanding

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

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Rishikesh
Posted 7 years ago. Direct link to Rishikesh's post “What about scalar divisio...”
What about scalar division, ie division of a matrix by a real number? Will the same rules apply as in scalar multiplication?
Button navigates to signup pageComment on Rishikesh's post “What about scalar divisio...”
(18 votes)
Answer
- Jasper Williams
  Posted 7 years ago. Direct link to Jasper Williams's post “Short answer - yes, Absol...”
  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)
  Button navigates to signup page
  (40 votes)
Violamaster
Posted 5 years ago. Direct link to Violamaster's post “what would a matrice mult...”
what would a matrice multiplied by a zero be?
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- kubleeka
  Posted 5 years ago. Direct link to kubleeka's post “You would get a matrix wh...”
  You would get a matrix where every entry is 0. This is called a zero matrix.
  Comment on kubleeka's post “You would get a matrix wh...”
  (22 votes)
Deepam Sarmah
Posted 7 years ago. Direct link to Deepam Sarmah's post “In the definition of scal...”
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?
Button navigates to signup pageComment on Deepam Sarmah's post “In the definition of scal...”
(11 votes)
Answer
allison.helton
Posted 4 years ago. Direct link to allison.helton's post “I was solving a Khan Acad...”
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.
Button navigates to signup pageComment on allison.helton's post “I was solving a Khan Acad...”
(7 votes)
Answer
- Nicolas
  Posted 3 years ago. Direct link to Nicolas's post “The Khan Academy problem ...”
  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.
  Button navigates to signup page
  (4 votes)
MoulikLaddha
Posted 3 years ago. Direct link to MoulikLaddha's post “what are scalar numbers?”
what are scalar numbers?
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(3 votes)
Answer
- obiwan kenobi
  Posted 3 years ago. Direct link to obiwan kenobi's post “Scalar are just your basi...”
  Scalar are just your basic numbers. 1, 2, 5, 1000, π are all scalars.
  Button navigates to signup page
  (8 votes)
ADITYA
Posted 6 years ago. Direct link to ADITYA's post “can anyone explain clearl...”
can anyone explain clearly with example on how to do fractional scalar division?
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(6 votes)
Answer
- Iza
  Posted a year ago. Direct link to Iza's post “With a fractional scalar ...”
  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.
  Button navigates to signup page
  (2 votes)
layaz7717
Posted 3 years ago. Direct link to layaz7717's post “So scalar division exists...”
So scalar division exists too, and we can use it, but we more often just multiply by the reciprocal, correct?
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(3 votes)
Answer
- loumast17
  Posted 3 years ago. Direct link to loumast17's post “That's kind of the defini...”
  That's kind of the definition of division, multiplying by the reciprocal. Or at least a definition. But yeah, you get it.
  Button navigates to signup page
  (3 votes)
dubilm23
Posted 21 days ago. Direct link to dubilm23's post “why.”
why.
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(4 votes)
Answer
zaitoumessaoud
Posted 3 years ago. Direct link to zaitoumessaoud's post “I cant solve the matrics ...”
I cant solve the matrics equations
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(3 votes)
Answer
- famousguy786
  Posted 3 years ago. Direct link to famousguy786's post “It took me a lot of time ...”
  It took me a lot of time too. Just keep practicing with full focus. You will get it.
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  (1 vote)
Mike
Posted 4 years ago. Direct link to Mike's post “I don't see how substitut...”
I don't see how substitution is helpful...it adds an extra step. You still have to do the work of merging the A and B matrix together in one way or another. 1/3X + A = B....why not just skip renaming them A and B and just subtract the left side matrix from the right side, then scale up by 3?
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(3 votes)
Answer