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## Properties of matrix multiplication

Current time:0:00Total duration:3:44

# Dimensions of identity matrix

## 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.