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## 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?

- at2:15it was middle column and top row.. isnt it?(12 votes)
- at2:01actually(3 votes)

- So we could factor a scalar from a matrix?(4 votes)
- 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!(8 votes)

- does the answer of negative numbers with scalar multiplication have to be +,or -?(4 votes)
- 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.(5 votes)

- Can we divide matrices ?(3 votes)
- 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.(5 votes)

- What does "scalar" mean?(3 votes)
- 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).(4 votes)

- can we raise a matrix to some power?(2 votes)
- 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.(4 votes)

- 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.(4 votes)

- 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?(2 votes) - What is the definition of a scalar? Can it be a decimal, or is it just an integer?(1 vote)
- A scalar can be any real number, so that includes integers, fractions, decimals, irrational numbers, etc.

It's because we're working with matrices that we use "scalar" to indicate that it is not a matrix, just a real number.(3 votes)

## Video transcript

Now that we know
what a matrix is, let's see if we
can start to define some operations on matrices. So let's say I have the 2 by 3
matrix, so two rows and three columns, and the entries are
7, 5, negative 10, 3, 8, and 0. And I want to define what
happens when I multiply 3 times this whole thing. So first of all, let's get
a little terminology out of the way. The number three, in
just the everyday world, if you weren't dealing
with matrices or vectors, and if you don't know
what vectors are, don't worry about
them just now, you would just call that a number. You would call
this a real number. It's just a regular
number sitting out there. But now in the
world where we have these new structured
things, these matrices, these arrays of
numbers, we will refer to these just plain
old real numbers that aren't part of some
type of an array here, we call these scalars. So essentially what
we're defining here, we don't know-- I haven't
said what this is actually going to turn out to be,
but whatever this turns out to be will be a product
of scalar multiplication, where we're multiplying
a scalar times a matrix. And so how would
you define this? What do you think
this should be? 3 times this stuff
right over here. Well, the world could have
defined scalar multiplication however it saw fit, but
one way that we find, perhaps, the most obvious
and the most useful, is to multiply this
scalar quantity times each of the entries. So this is going to
be equal to 3 times 7 in the top left, 3 times
5, 3 times negative 10, 3 times 3, 3 times
8, and 3 times 0, which will give us--
it didn't change the dimensions of the matrix. It didn't change,
I guess you could say, the structure
of the matrix, it just multiplied each
of the entries times 3. So the top left
entry is now going to be 21, the entry
in the middle row, top column is going to be
15, negative 30, 9, 24 and 0. So when you multiply a
matrix times a scalar, you just multiply each of
those entries times that scalar quantity.