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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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# Dimensions of identity matrix

Sal explains why the identity matrix is always a square matrix, even though it works with non-square matrices. Created by Sal Khan.

## Want to join the conversation?

- I dont understand. I thought IC=CI. But if IC=CI then in this case CI is not defined. Because the number of columns of C is not the same as the number of rows of I.(17 votes)
- IC = C

CI = C

If C is a square matrix, then the dimensions of the identity matrices in IC and CI are the same.

If C isn't a square matrix, then the dimensions of the identity matrices in IC and CI will be different but both equations will still be valid.(45 votes)

- Just curious, is the identity matrix always going to be a square matricies? I'm not sure how it would work if it wasn't a square matricies (if that's possible). Thanks.(15 votes)
- Yes, the identity matrix is always nxn, or square.

Great intuition.

Keep Studying!(16 votes)

- What is a 1x1 identity matrix?(3 votes)
- Usually we don't write 1X1's as matrices. We just call them numbers. But [1] would work for your purposes.(30 votes)

- Is the Identity Matrix commutative?(4 votes)
- Yes it is.

It and the zero matrix are the only ones that are.

Let Z be an nxn zero matrix,

Let I be an nxn identity matrix

Let A be an nxn matrix

then

AI = IA = A

AZ = ZA = 0(3 votes)

- Just to make it clear when i get a MCQ on this one. So the commutative property does work in matrices but not in all cases. right??(3 votes)
- The commutative property
work with matrices. This is obvious with matrices that have dissimilar dimensions, but it is also true for square, or nxn matrices.**does not**

Matrices are part of a "family", (we use the description "algebraic ring" in math) that have the particular property that they are non commutative (most rings*are*commutative, such as the real numbers, which is also are a ring).

What it means to be a non-commutative ring is that there exists an*a and b*such that*a·b≠b·a*.

It is a rare case that*a·b=b·a*for any*a or b*that is not the zero matrix nor the identity matrix.

So, the only time that commutivity is assured with matrices is when at least one of*a or b*is the zero matrix or is the identity matrix.**The bottom line regarding commutivity and matrices is exactly the opposite of what you stated.**

The commutative property DOES NOT work with matrices except in the case of the identity and/or zero matrix(4 votes)

- How can we apply identity matrix? Is it designed to simplify any process or calculation? Thanks.(4 votes)
- Its just good to know what 1 is and really get use to the properties of matrices. Using this basic concept you can know how to adjust numbers to make other matrix you encounter easily to comprehend and summarize. As well as you can practice some essential steps such as rows and column rules.(1 vote)

- What would be the dimensions of the product CxI, given that dimensions of C are axb in which a and b are different values ? In this case would the dimensions of I be bxb ?(3 votes)
- The product CI = C. Hence, the dimensions of the product CI is axb.

Yes, the dimensions of I would be bxb.(2 votes)

- If matrix A is non-squared, then the identity matrix in the equation IC = C is different from the identity matrix in CI = C... correct? But ... then CI DOES NOT equal IC?? Because you changed the I? Similarly, AB does not equal BA, if you suddenly say that the B is now something different than before? So am I missing something, or is IC = CI only valid for squared matrices?(3 votes)
- Yes, the left and right identity matrices are not the same size, and are therefore not the same matrix. Let's say I_l (capital i underscore lowercase L) is the left identity matrix, and that I_r is the right identity matrix.

I_l A = A I_r is valid for all matrices, since by definition, I_l A = A and A I_r = A. The fact that matrix multiplication isn't commutative is not relevant here.(1 vote)

- 3b1b's essense of Linear Algebra was really superb

I understood a lot of the optics in matrices including these in a really intuitive way.

The reason the I matrix is always square is because (in this case, where I is on the left), it is used as a transformation matrix.

The matrix on the right can be treated as vector with n dimensions where n is the number of rows on that 2nd matrix. The I matrix has to match the number the number of dimensions of that 2nd matrix then by matching the number of rows through its columns.

The reason they're square though is because they are BASIS VECTORS where none of the vectors are linearly dependent. In order for that to happen, they must live in different dimensions. So the number of dimensions of that I matrix is the same as its number of columns.

But we already know that number of col = num of rows of the 2nd matrix. Therefore the I matrix would be n*n where n=num of of of the 2nd matrix.(2 votes) - Do we agree that, in this example, if we take CI=C then I would be a bxb square matrix ?(2 votes)
- That is correct. The identity matrix used as a multiplier is an a x a square matrix and the identity matrix as the multiplicand is a b x b square matrix.(1 vote)

## Video transcript

Voiceover:Let's say that
you've got some matrix C, trying my best to bold it, to make sure you realize
that this is a matrix. Let's say that we know that it has a rows and b columns. It's an a by b matrix. Let's say that we are going to multiply it by some identity matrix. Once again let me do my
best to attempt to bold this right over here. We're going to multiply the
identity matrix I times C and of course we are going to get C again because that's the identity matrix, that's the property of
the identity matrix. Of course C, we already
know is an a by b matrix. A rows and b columns. Based on this, what
are the dimensions of I going to be? I encourage you to pause
this video and think about it on your own. We've already done this
exercise a little bit, where we first looked at identity matrices but now we're doing it with a very ... We're multiplying the identity matrix times a very general matrix. I'm just even speaking in generalities about these dimensions. Well one thing we know is
that matrix multiplication is only defined is if the column, the number of columns of the first matrix is
equal to the number of rows of the second matrix. This one has a rows, so this
one's going to have a columns. Now how many rows is
this one going to have? We already know that matrix
multiplication is only defined if the number of columns
on the first matrix is equal to the number of
rows on the second one. We know that the product
gets its number of rows from the number of rows of the first matrix being multiplied. The product has a rows then the identity matrix right over here has to have a rows. What's interesting about this? When we first got introduced
to identity matrices, we were multiplying, we picked out a three by three example and we got a three by
three identity matrix. What's interesting about what
we've just proven to ourselves is the identity matrix for any matrix, even a non square matrix, a and b could be two different values. The identity matrix for any matrix is going to be a square matrix. It's going to have the same number of rows and the same number of columns. When we think about identity matrices, we can really just say, well
is this the identity matrix that is a four by four? Is it a three by three? Is it a two by two? Or I guess one by one? The convention is, it isn't
even to write identity two by two is equal to
one, zero, zero, one. The convention is actually just write I2 because you know it's
going to be a two by two. It's going to be a two by two matrix, it's going to be one, zero, zero, one. Identity five is going to
be a five by five matrix. It's going to be one,
one, two, three, four. Zero, one, two, one, three. Zero, zero, one, zero, zero. Zero ... you get the idea, zero, zero, zero, one, zero. Zero, zero, zero, zero, one. Just like that. The whole point here is just to realize that your identity matrix is always going to be a square matrix and it works even when you're multiplying non square other matrices.