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## Precalculus

### Course: Precalculus>Unit 7

Lesson 3: Multiplying matrices by scalars

# Multiplying matrices by scalars

Sal defines what it means to multiply a matrix by a scalar (in the world of matrices, a scalar is simply a regular number). Created by Sal Khan.

## Want to join the conversation?

• at it was middle column and top row.. isnt it?
• obviously a mistake!i've noticed that as well but didnt budge cause it doesnt ruin our understandings
(1 vote)
• So we could factor a scalar from a matrix?
• Yep! If I have the matrix
╔ 2 10 14 ╗
║ 6 42 12 ║
╚ 20 4 8 ╝,
then I can factor a 2 out:
...╔ 1 5 7 ╗
2║ 3 21 6 ║
..╚ 10 2 4 ╝
(The dots are there because the matrix did not appear correctly when I did spaces... It must be a Khan Academy bug.)

Hope this helps!
• What does "scalar" mean?
• In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector.

The term "scalar" itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is the multiplication of a vector by a scalar (where the product is a vector), and must be distinguished from inner product of two vectors (where the product is a scalar).
• does the answer of negative numbers with scalar multiplication have to be +,or -?
• The answer for each multiplication of the scalar times the item in the matrix being multiplied has to follow the rules of signed numbers. In other words, a negative times a negative results in a positive, while a positive times a negative results in a negative result.

If you multiply the matrix [8 0 -3] times -5 as shown below
-5 ∙ [8 0 -3]
you get [(-5∙8) (-5∙0) (-5∙ -3)]
=[-40 0 15]

If you multiplied the same matrix times +5
5 ∙ [8 0 -3]
you get [(5∙8) (5∙0) (5∙ -3)]
=[40 0 -15]

Keeping track of the negatives is just one of the fun challenges of working with matrices.
• I am trying to solve a problem using scalar math to try and find values of x and y. KHAN article does not specifically state how to find values of x and y just how to work the problem . It does not get specific on how to find x and y
• Hi Gregory,
You have probably figured this out by now, but if not, or for anyone else who (like me) was thrown off by this, what I think is confusing about the exercise is the format. It is showing a matrix being multiplied by a scalar, then showing what the results of that multiplication are.

Here's an example: 4x [x 4 5] = [12 16 y]
You are supposed to figure out what x (the number being multiplied) is for element 1,1, and then what y (the product is) for element 1,3.

So to solve for x you would just need to set up:
4x=12
And for y:
4(5)=y
• Can we divide matrices ?
• Not generally, no. Some matrices have an inverse, which is the matrix you multiply them by to get the identity matrix. So in that case, you can multiply by the inverse, which is like dividing.

But if you have matrices A, B, C, A has no inverse, and AB=AC, then it's not necessarily the case that B=C.
• can we raise a matrix to some power?
• Short answe, yes.

Long answer, raising to a power is multiplying something by itself. So for a matrix to be able to be multiplied by itself, it needs the same number of rows and columns. So it needs to be a square matrix. So as long as you have a square matrix you can raise it to a power.
• Can matrix also multiply with matrix?
(1 vote)
• Can we multiply a matrix by a negative fraction? Would this work the same way?
(1 vote)
• Yes, you can multiply a matrix by any scalar, there are no limitations, and the operation works exactly the same.
• Are you allowed to use scalars to find matrix inverses?

Ex.
| 1 0 0 : -7 6 4 |
| 0 4 0 : 28 -20 -12 |
| 0 0 1 : -3 2 1 |

Then can you multiply the second row by 1/4 to get the inverse?