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In this video we'll talk about

matrix addition and subtraction,

as well as how to

multiply a matrix by a

number, also called Scalar Multiplication.

Let's start an example.

Given two matrices like these,

let's say I want to add them together.

How do I do that?

And so, what does addition of matrices mean?

It turns out that if you

want to add two matrices, what

you do is you just add

up the elements of these matrices one at a time.

So, my result of adding

two matrices is going to

be itself another matrix and

the first element again just by

taking one and four and

multiplying them and adding them together, so I get five.

The second element I get

by taking two and two

and adding them, so I get

four; three plus three

plus zero is three, and so on.

I'm going to stop changing colors, I guess.

And, on the right is open

five, ten and two.

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because this has 3

rows and 2 columns, so it's 3 by 2.

This is also a 3

by 2 matrix, and the

result of adding these two

matrices is a 3 by 2 matrix again.

So you can only add

matrices of the same

dimension, and the result

will be another matrix that's of

the same dimension as the ones you just added.

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Where as in contrast, if you

were to take these two matrices, so this

one is a 3 by

2 matrix, okay, 3 rows, 2 columns.

This here is a 2 by 2 matrix.

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And because these two matrices

are not of the same dimension,

you know, this is an error,

so you cannot add these

two matrices and, you know,

their sum is not well-defined.

So that's matrix addition.

Next, let's talk about multiplying matrices by a scalar number.

And the scalar is just a,

maybe a overly fancy term for,

you know, a number or a real number.

Alright, this means real number.

So let's take the number 3 and multiply it by this matrix.

And if you do that, the result is pretty much what you'll expect.

You just take your elements

of the matrix and multiply

them by 3, one at a time.

So, you know, one

times three is three.

What, two times three is

six, 3 times 3

is 9, and let's see, I'm

going to stop changing colors again.

Zero times 3 is zero.

Three times 5 is 15, and 3 times 1 is three.

And so this matrix is the

result of multiplying that matrix on the left by 3.

And you notice, again,

this is a 3 by 2

matrix and the result is

a matrix of the same dimension.

This is a 3 by

2, both of these are

3 by 2 dimensional matrices.

And by the way,

you can write multiplication, you know, either way.

So, I have three times this matrix.

I could also have written this

matrix and 0, 2, 5, 3, 1, right.

I just copied this matrix over to the right.

I can also take this matrix and multiply this by three.

So whether it's you know, 3

times the matrix or the

matrix times three is

the same thing and this thing here in the middle is the result.

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You can also take a matrix and divide it by a number.

So, turns out taking

this matrix and dividing it by

four, this is actually the

same as taking the number

one quarter, and multiplying it by this matrix.

4, 0, 6, 3 and

so, you can figure

the answer, the result of

this product is, one quarter

times four is one, one quarter times zero is zero.

One quarter times six is,

what, three halves, about six over

four is three halves, and

one quarter times three is three quarters.

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And so that's the results

of computing this matrix divided by four.

Vectors give you the result.

Finally, for a slightly

more complicated example, you can

also take these operations and combine them together.

So in this calculation, I

have three times a vector

plus a vector minus another vector divided by three.

So just make sure we know where these are, right.

This multiplication.

This is an example of

scalar multiplication because I am taking three and multiplying it.

And this is, you know, another

scalar multiplication.

Or more like scalar division, I guess.

It really just means one zero times this.

And so if we evaluate

these two operations first, then

what we get is this thing

is equal to, let's see,

so three times that vector is three,

twelve, six, plus

my vector in the middle which

is a 005 minus

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one, zero, two-thirds, right?

And again, just to make

sure we understand what is going on here,

this plus symbol, that is

matrix addition, right?

I really, since these are

vectors, remember, vectors are special cases of matrices, right?

This, you can also call

this vector addition This

minus sign here, this is

again a matrix subtraction,

but because this is an

n by 1, really a three

by one matrix, that this

is actually a vector, so this is

also vector, this column.

We call this matrix a vector subtraction, as well.

OK?

And finally to wrap this up.

This therefore gives me a

vector, whose first element is

going to be 3+0-1,

so that's 3-1, which is 2.

The second element is 12+0-0, which is 12.

And the third element

of this is, what, 6+5-(2/3),

which is 11-(2/3), so

that's 10 and one-third

and see, you close this square bracket.

And so this gives me a

3 by 1 matrix, which is

also just called a 3

dimensional vector, which

is the outcome of this calculation over here.

So that's how you

add and subtract matrices and

vectors and multiply them by scalars or by row numbers.

So far I have only talked

about how to multiply matrices and

vectors by scalars, by row numbers.

In the next video we will

talk about a much more

interesting step, of taking

2 matrices and multiplying 2 matrices together.