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## Properties of matrix multiplication

# Defined matrix operations

## Video transcript

So we have matrix D and matrix
B and they ask us is DB defined? Is the product D
times B defined? So D times B is
going to be defined is if-- let me make
this very clear. This is how I think about it. So let me copy and
paste this so I can do this on my scratch pad. So to answer that
question, get out the scratch pad right over here. Let me paste the
question right here. So let's think about
these two matrices. You first have matrix D. I'll
do this a nice bold D here. And it has three rows
and three columns. So it is a 3 by 3 matrix. And then you'll want to multiply
that times matrix B. Matrix B is a 2 by 2 matrix. The only way that we know to
define matrix multiplication is if these middle two
numbers are the same. If the number of columns D has
is equal to the number of rows B has. Now in this case, they clearly
do not equal each other, so matrix multiplication
is not defined here. So let's go back
there and say no. No, DB is not defined. Let's do a few more
of these examples. So then we have a 2 by 1,
you could view this is as a 2 by 1 matrix or you could
view this as a column vector. This is another 2 by 1
matrix, or a column vector. Is C plus B defined. Well, matrix addition is
defined if both matrices have the exact same dimensions,
and these two matrices do have the exact
same dimensions. And the reason why is
because with matrix addition, you just add every
corresponding term. So in the sum the
top, it'll actually be 4 plus 0 over negative 2 plus
0, which is still just going to be the same thing
as this matrix up here. But what they're
asking is this defined? Absolutely, these both
are 2 by 1 matrices, so yes, it is defined. Let's do one more. So once again,
they're asking us is the product A times E defined? So here you have
a 2 by 2 matrix. Let me copy and paste this
just so we can make sure that we know what
we're talking about. So get my scratch pad out. So this top matrix right
over here, so matrix A is a 2 by 2 matrix. And matrix E, so we're
going to multiply it times matrix E, which has
one row and two columns. So in this scenario
once again, the number of rows-- sorry-- the number
of columns matrix A has is two and the number of rows
matrix E has is one, so this will not be defined. These two things have to be the
same for them to be defined. Now, what is interesting is,
if you did it the other way around, if you took E
times A, let's check if this would have been defined. Matrix E is 1 by 2, one
row times two columns. Matrix A is a 2 by 2,
two rows and two columns, and so this would
have been defined. Matrix E has two
columns, which is exactly the same number of
rows that matrix A has. And this really
hits the point home that the order matters
when you multiply matrices. But for the sake of this
question, is AE defined? No, it isn't. And so we can check our
answer, no, it isn't.