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## Precalculus

### Course: Precalculus > Unit 7

Lesson 11: Properties of matrix multiplication- Defined matrix operations
- Matrix multiplication dimensions
- Intro to identity matrix
- Intro to identity matrices
- Dimensions of identity matrix
- Is matrix multiplication commutative?
- Associative property of matrix multiplication
- Zero matrix & matrix multiplication
- Properties of matrix multiplication
- Using properties of matrix operations
- Using identity & zero matrices

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# Zero matrix & matrix multiplication

Just as any number multiplied by zero is zero, there is a zero

*such that any matrix multiplied by it results in that zero matrix. Learn more from Sal. Created by Sal Khan.***matrix**## Want to join the conversation?

- Couldn't you just multiply the matrix by 0?(34 votes)
- That would make a zero matrix indeed. It is one and the same thing.(29 votes)

- There are multiple zero matrices that you could use. So the zero matrix is not unique, right?

E.g. in the last example any matrix of zeroes with 3 rows would work.(7 votes)- Interesting ... but not quite right ... for the last example any zero matrix with 2 columns can be used. The number of rows will dictate the number of rows in the result --

e.g.s:`[0 0] * [1 2 3] = [0 0 0]`

[4 5 6]

[0 0] * [1 2 3] = [0 0 0]

[0 0] [4 5 6] [0 0 0]

[0 0] * [1 2 3] = [0 0 0]

[0 0] [4 5 6] [0 0 0]

[0 0] [0 0 0](13 votes)

- For a given matrix A with format
`NxM`

, does the zero matrix for A needs to be**strictly**a`MxN`

matrix filled with zeros? I think the answer is yes, so it can satisfy`0 x A = A x 0 = 0`

But what about a matrix that satisfies`0 x A = 0`

but do**not**satisfies`A x 0 = 0`

? A`PxN`

matrix filled with zeros where`P != M`

is one example. Is it an*asymmetric zero matrix*or something like that?(4 votes)- Sal mentions that when he writes
**0**as a zero matrix, it may have different dimensions depending on what it is multiplying, or being multiplied by. This is even "worse" than the identity matrix**I**, which is always square, and whose dimension is determined by the other matrix in the multiplication. If A is

we can multiply it (on the left) by`NxM`

**any**

matrix full of 0's to get a`JxN`

result that is also full of 0's, and similarly for multiplication on the right. So when you say`JxM`

or`I x A = A`

, at least you know the dimension(s) of the identity matrix(es) if you know the dimension of A. When you say`A x I = A`

, the`0 x A = 0`

on the left and the`0`

on the right may not be the same, and aren't even uniquely determined.`0`

(9 votes)

- btw........ is "Zero" matrix and "Null" matrix the same thing ?(4 votes)
- Yes - the zero matrix and the null matrix are two names for the same thing.(5 votes)

- is a 3 by 3 zero matrix equivalent to a 2 by 2 zero matrix?(3 votes)
- Well, they are both zero matrices, but they are not interchangeable. If you want to multiply a 2 x 3 matrix C by a zero matrix Z in the order ZC, you would need the

2 x 2 zero matrix. The result would be a 2 x 3 zero matrix.

The 3 x 3 zero matrix multiplied in the order ZC would be undefined.

On the other hand if you want to multiply the same matrix C by a zero matrix Z in the order CZ, you would need the 3 x 3 zero matrix. The result would be a 2 x 3 zero matrix. If you tried to multiply the order CZ using the 2 x 2 zero matrix that worked for ZC, the result would now be undefined.

By the way, you could also use a 5 x 2 zero matrix in the order ZC. The result would be a 5 x 3 zero matrix.

Hope that helps.(5 votes)

- Can you raise a matrix to the power of n?(4 votes)
- Yes, for integer n. A^n is just n copies of the matrix A multiplied together.(3 votes)

- Where did sal say that I(A) = A = A(I) = A? I think I might have missed that. Or is that a mistake? Is it really true that any matrix (first) times an Identity Matrix(second) equals the first matrix?(2 votes)
- In the first video of this section, "Identity matrix", Sal has shown that IA=A, and then at the end of the video he encouraged us viewers to test AI=A on our own.(4 votes)

- At3:23, in the question you gave us, you multiplied the matrix (containing elements 1,2,3,4,5,6) with the zero matrix (with dimensions 3x2). Now my question is, cant the zero matrix have its dimensions as nx2? Because the matrix multiplication would still be valid, right? If so, then the matrix product would have dimensions as nx3. Therefore, unlike inverse matrix, which is unique for every matrix, there can be 'n' number of zero matrices for a given matrix. Am I correct?(2 votes)
- Yes. For a nxm matrix A, and zero matrix Z, both AZ and ZA are defined for an infinite number of zero matrices.(3 votes)

- Can you multiply matrices with exponents, put them into equations, or use them for variables?(3 votes)
- you should be able too, if you watch later videos on matrices sal might talk about it IDK for sure tho(1 vote)

- Is there any reason why a zero matrix wouldn't always be a square matrix like the identity matrix is? Wouldn't you get a matrix of the same dimensions as the original that's all zeros if you multiplied it by a square zero matrix? (say in the example in the video, why not multiply by a 2x2 zero matrix instead of a 2x3?)(2 votes)
- 1. A zero matrix is just a matrix with any dimensions that has all elements inside the matrix as 0. It does NOT have to be a square matrix.

2. You are right. Sal could have multiplied a 2x2 zero matrix with the 2x3 matrix to obtain a resulting zero matrix. Having a 2x3 zero matrix makes no difference as having a 3x3 matrix. They are both zero matrices.(3 votes)

## Video transcript

Voiceover:We've been drawing analogies between I guess we could say traditional multiplication,
or scalar multiplication, and the first one we drew is when you have traditional multiplication, you multiply 1 times any number and you get that number again. And you could view 1 as
essentially the identity. The identity number, or this is the identity
property of multiplication. You multiply 1 times any number, you get that number again. And that essentially inspired our thinking behind having identity matrices. Said hey, maybe there are some matrices that if I multiply
times some other matrix, I'm going to get that matrix again. And you've probably shown for yourselves that you can do it in either way. You could have some matrix
times an identity matrix and get that matrix again. Now if matrix A right over
here is a square matrix, then in either situation,
this identity matrix is going to be the same identity matrix. But if matrix A is not a square matrix, then these are going to be two different identity matrices, depending on the appropriate dimensions. Now, let's see if we
can extend this analogy between traditional multiplication and matrix multiplication. We know that there's
another spacial number in traditional multiplication,
and that's a 0. So, we know that 0 times
anything is equal to 0. Or, anything times 0 is equal to 0. So what would be the analogy if we're thinking about
matrix multiplication? Well, it would be some matrix that if I were to multiply
it times another matrix, I get, I guess you could say
that same 0 matrix again. And that is what we call it. We call it a 0 matrix. So if I take some matrix
A, and essentially, if I multiply it times
one of these 0 matrices, or I multiply one of
the 0 matrices times A, I should get another 0 matrix. And it depends on the dimensions. You might not get a 0 matrix with the same dimensions. It depends what the dimensions of A are going to be, but you could image what a 0 matrix might look like. For example, if A is 1,2,3,4, what's a 0 matrix that I could multiply this by to get another 0 matrix? Well, it might be pretty straight forward, if you just had a ton of zeros here, when you multiply this out, you're going to get this - you date the dot product of
this row and this column. 0 times 1 plus 0 times 3 is going to be 0. You keep going, 0,0,0,0. If we had a - just to
make the point clear - let's say we had a matrix 1,2,3,4,5,6. So over here, we want
to multiply this times - let's see, in order for
the matrix multiplication to work, my 0 matrix has got to have the same number of columns
as this one has rows, so it's got to have 2 columns, but I could make it have 3 rows. So it could look like this, 0,0,0,0 and I encourage you to multiply these two. Pause the video right
now, and see what you get. Well when you multiply
them, let's think about it. So the top left entry - so let me just write the dimensions. This is a 3x2 matrix, this is a 2x3matrix. So, we know that we have
valid matrix multiplication going on right over same here. The number of columns in the first matrix is equal to the number of
rows in the second one. And we also know that
the resulting product is going to be a 3x3 matrix. So it's going to be a 3x3 matrix, and I'll leave it up to you to verify that all the entries
here are going to be 0. And it makes sense, you
could go through the math, but you can see, well
you're just everytime, you're multiplying say
this row by this column to get that entry, we'll just have 0 times 1 plus 0 times
4 to get that 0 there. But the whole point of
showing you this example is, we have one 0 matrix multiplying by this matrix right over here, and then we get another 0 matrix but it has different dimensions.