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Matrix multiplication is really

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useful since you can pack

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a lot of computation into just

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one matrix multiplication operation.

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But you should be careful of how you use them.

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In this video I want to

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tell you about a few properties of matrix multiplication.

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When working with just raw

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numbers or when working with

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scalars, multiplication is commutative.

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And what I mean by that is

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if you take three times

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five, that is equal

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to five times three and

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the ordering of this multiplication doesn't matter.

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And this is called the commutative

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property of multiplication of real numbers.

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It turns out this property that

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you can, you know, reverse

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the order in which you

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multiply things, this is not true for matrix multiplication.So

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concretely, if A and

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B are matrices, then in

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general, A times B is

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not equal to B times

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A.  So just be careful of that.

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It's not okay to arbitrarily reverse

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the order in which you are multiplying matrices.

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So, we say that matrix multiplication

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is not commutative, it's a fancy

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way of saying it.

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As a concrete example, here

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are two matrices, matrix 1100

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times 0020, and if you multiply

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these two matrices, you get this result on the right.

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

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So, I'm going to take these

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two matrices and just reverse them.

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It turns out if you multiply

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these two matrices, you get

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the second answer on the

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right and, you know, real

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clearly, these two matrices are

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not equal to each other.

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So, in fact, in

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general, if you have

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a matrix operation like

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A times B. If A

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is an m by n matrix

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and B is an by

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M matrix, just as an example.

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Then, it turns out

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that the matrix A times

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B right, is going

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to be an m by

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m matrix, where as

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the matrix b x a

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is going to be an n

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by n matrix so the

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dimensions don't even match, right,

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so A times B and

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B times A may not even be the same dimension.

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In the example on the left,

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I have all two by two matrices,

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so the dimensions were the same,

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but in general reversing the

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order of the matrices

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can even change the dimension

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of the outcome so

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matrix multiplication is not commutative.

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Here's the next I want to talk about.

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So, when talking about real

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numbers, or scalars, let's

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see, I have 3 times 5 times 2.

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I can either multiply 5

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times 2 first, and

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I can compute this as 3 times 10.

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Or, I can multiply

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three times five for us and

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I can compute this as, you

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know fifteen times two and

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both of these give you the same answer, right?

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Each, both of these is equal

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to thirty so Whether I

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multiply five times

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two first or whether I

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multiply three times five

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first because well, three

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times five times two

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is equal to three times

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five times two.

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And this is called the

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associative property of role number multiplication.

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It turns out that matrix multiplication is associative.

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So concretely, let's say

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I have a product of three

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matrices, A times B times

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C. Then I can

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compute this either as A

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times, B times C

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or I can compute this as

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A times B, times C

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and these will actually give me the same answer.

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I'm not going to prove this, but

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you can just take my word for it, I guess.

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So just be clear what I mean by

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these two cases, let's look

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at first one first case.

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What I mean by that is

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if you actually want to compute

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A times B times C, what

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you can do is you can

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first compute B times C.

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So that D equals B time

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C, then compute A times

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D.  And so this is really

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computing a times B

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times C.  Or, for

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this second case, You can

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compute this as, you can

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set E equals A

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times B.  Then compute E

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times C.  And this

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is then the same as a

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times B times C

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and it turns out that both

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of these options will give

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you, is guaranteed to give you the same answer.

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And so we say that matrix

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multiplication does enjoy the associative property.

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Okay?

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And don't worry about the terminology

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associative and commutative that's

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why there's not really going to use

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this terminology later in these

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

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Finally, I want to

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tell you about the identity

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matrix, which is special matrix.

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So let's again make the

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analogy to what we know

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of raw numbers, so when dealing

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with raw numbers or scalar

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numbers, the number one,

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is you can think

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of it as the identity of multiplication,

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and what I mean by that

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is for any number

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Z, the number 1

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times z is equal

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to z times one, and

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that's just equal to

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the number z, right, for any raw number.

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Z. So 1 is

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the identity operation and so it satisfies this equation.

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So it turns out that

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in the space of matrices as

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an identity matrix as well.

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And it's unusually denoted i,

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or sometimes we write it

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as i of n by

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n we want to make explicit the dimensions.

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So I subscript n by n

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is the n by n identity matrix.

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And so there's a different identity

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matrix for each dimension n and are a few examples.

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Here's the two by two identity

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matrix, here's the three

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by three identity matrix, here's the four by four identity matrix.

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So the identity matrix, has the

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property that it has

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ones along the diagonals,

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right, and so on and

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is zero everywhere else, and

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so, by the way the

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one by one identity matrix is just a number one.

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This is one by one matrix

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just and it's not a very

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interesting identity matrix and informally

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when I or others are being

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sloppy, very often, we will

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write the identity matrix using fine notation.

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I draw, you know, let's

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go back to it and just write 1111,

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dot, dot, dot, 1

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and then we'll, maybe, somewhat

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sloppily write a bunch of zeros there.

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And these zeros, this

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big zero, this big zero

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that's meant to denote that this

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matrix is zero everywhere except for

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the diagonals, so this is just

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how I might sloppily write

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this identity matrix

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She says property that for

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any matrix A, A times

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identity i times A

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A. So that's a lot

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like this equation that we have up here.

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One times z equals z times

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one, equals z itself so

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I times A equals A

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times I equals A.  Just

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make sure we have the dimensions right, so

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if A is a n

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by n matrix, then this

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identity matrix that's an

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m by n identity matrix.

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And if A is m by

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n then this identity

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matrix, right, for matrix

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multiplication make sense that has a

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m by n matrix because

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this m has a match

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up that m And

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in either case the outcome

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of this process is you

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get back to Matrix A, which

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is m by n.

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So whenever we write

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the identity matrix I, you

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know, very often the dimension rightwill

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be implicit from the context.

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So these two I's they' re

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actually different dimension matrices, one

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may be N by N, the other

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is M by M But when

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we want to make the dimension

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of the matrix explicit, then sometimes

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we'll write to this I subscript

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N by N, kind of like we have up here.

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But very often the dimension will be implicit.

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Finally, just want to point

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out that earlier I

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said that A times B

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is not in general equal

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to B times A, right?

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That for most matrices A and B, this is not true.

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

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That A times the identity

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matrix does indeed equal to

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identity times A, it's

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just that this is not true

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for other matrices, B in general.

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So that's it for the

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properties of matrix multiplication.

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And the special matrices, like the

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identity matrix I want to

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tell you about, in the next

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and final video now linear algebra review.

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I am going to quickly tell you

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about a couple of special

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matrix operations, and after

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that you know everything you need

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to know about linear algebra for this course