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When working with just real numbers or

when working with scalars, multiplication is commutative.

And what I mean by that is that if you take 3 times 5,

that is equal to 5 times 3.

And the ordering of this multiplication doesn't matter.

And this is called the commutative property

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of multiplication of real numbers.

It turns out this property,

they can reverse the order in which you multiply things.

This is not true for matrix multiplication.

So concretely, if A and B are matrices.

Then in general, A times B is not equal to B times A.

So, just be careful of that.

Its not okay to arbitrarily reverse the order in which you multiply matrices.

Matrix multiplication in not commutative, is the fancy way of saying it.

As a concrete example, here are two matrices.

This matrix 1 1 0 0 times 0 0 2 0 and

if you multiply these two matrices you get this result on the right.

Now let's swap around the order of these two matrices.

So I'm gonna take this two matrices and just reverse them.

It turns out if you multiply these two matrices,

you get the second answer on the right.

And well clearly, right, these two matrices are not equal to each other.

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So, in fact, in general if you have a matrix operation like A times B,

if A is an m by n matrix,

and B is an n by m matrix, just as an example.

Then, it turns out that the matrix A times B,

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right, is going to be an m by m matrix.

Whereas the matrix B times A is going to be an n by n matrix.

So the dimensions don't even match, right?

So if A x B and B x A may not even be the same dimension.

In the example on the left, I have all two by two matrices.

So the dimensions were the same, but in general, reversing the order of

the matrices can even change the dimension of the outcome.

So, matrix multiplication is not commutative.

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Here's the next property I want to talk about.

So, when talking about real numbers or scalars, let's say I have 3 x 5 x 2.

I can either multiply 5 x 2 first.

Then I can compute this as 3 x 10.

Or, I can multiply 3 x 5 first, and I can compute this as 15 x 2.

And both of these give you the same answer, right?

Both of these is equal to 30.

So it doesn't matter whether I multiply 5 x 2 first or

whether I multiply 3 x 5 first, because sort of,

well, 3 x (5 x 2) = (3 x 5) x 2.

And this is called the associative property of real number multiplication.

It turns out that matrix multiplication is associative.

So concretely, let's say I have a product of three matrices A x B x C.

Then, I can compute this either as A x (B x C) or

I can computer this as (A x B) x C,

and these will actually give me the same answer.

I'm not gonna prove this but you can just take my word for it I guess.

So just be clear, what I mean by these two cases.

Let's look at the first one, right.

This first case.

What I mean by that is if you actually wanna compute A x B x C.

What you can do is you can first compute B x C.

So that D = B x C then compute A x D.

And so this here is really computing A x B x C.

Or, for this second case, you can compute this as,

you can set E = A x B, then compute E times C.

And this is then the same as A x B x C, and it turns out that

both of these options will give you this guarantee to give you the same answer.

And so we say that matrix multiplication thus enjoy the associative property.

Okay?

And don't worry about the terminology associative and commutative.

That's what it's called, but I'm not really going to use this terminology later

in this class, so don't worry about memorizing those terms.

Finally, I want to tell you about the Identity Matrix,

which is a special matrix.

So let's again make the analogy to what we know of real numbers.

When dealing with real numbers or scalar numbers, the number 1,

you can think of it as the identity of multiplication.

And what I mean by that is that for

any number z, 1 x z = z x 1.

And that's just equal to the number z for any real number z.

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So 1 is the identity operation and so it satisfies this equation.

So it turns out, that this in the space of matrices there's

an identity matrix as well and it's usually denoted I or

sometimes we write it as I of n x n if we want to make it explicit to dimensions.

So I subscript n x n is the n x n identity matrix.

And so that's a different identity matrix for each dimension n.

And here are few examples.

Here's the 2 x 2 identity matrix, here's the 3 x 3 identity matrix,

here's the 4 x 4 matrix.

So the identity matrix has the property that it has ones along the diagonals.

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All right, and so on.

And 0 everywhere else.

And so, by the way, the 1 x 1 identity matrix is just a number 1,

and so the 1 x 1 matrix with just 1 in it.

So it's not a very interesting identity matrix.

And informally, when I or others are being sloppy,

very often we'll write the identity matrices in fine notation.

We'll draw square brackets, just write one one one dot dot dot dot one, and

then we'll maybe somewhat sloppily write a bunch of zeros there.

And these zeroes on the, this big zero and this big zero,

that's meant to denote that this matrix is zero everywhere except for the diagonal.

So this is just how I might swap you the right D identity matrix.

And it turns out that the identity matrix has its property that for

any matrix A, A times identity equals I times A equals A so

that's a lot like this equation that we have up here.

Right?

So 1 times z equals z times 1 equals z itself.

So I times A equals A times I equals A.

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So whenever we write the identity matrix I, you know,

very often the dimension Mention, right, will be implicit from the content.

So these two I's, they're actually different dimension matrices.

One may be n by n, the other is n by m.

But when we want to make the dimension of the matrix explicit, then

sometimes we'll write to this I subscript n by n, kind of like we had up here.

But very often, the dimension will be implicit.

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Finally, I just wanna point out that earlier

I said that AB is not, in general, equal to BA.

Right?

For most matrices A and B, this is not true.

But when B is the identity matrix, this does hold true,

that A times the identity matrix does indeed equal to identity

times A is just that you know this is not true for other matrices B in general.