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.