### Video Transcript

Suppose the matrix 𝐴 is the one-by
three matrix one, negative two, negative three and the matrix 𝐵 is the three-by-one
matrix eight, one, negative three. Find the product 𝐴𝐵 and find the
product 𝐵𝐴.

So before we start to answer this
question, let’s remind ourselves a little bit about matrix multiplication. If we’ve got the matrix 𝐴
multiplied by the matrix 𝐵, well then, first of all, what we can think about is if
we have, for instance, an 𝑚-by-𝑛 matrix multiplied by an 𝑛-by-𝑝 matrix, to
enable our multiplication to happen, then one thing must be true. And that is that the second
dimension of our first matrix must be the same as the first dimension of our second
matrix. So in this case, we can see we’ve
got the shared 𝑛. And then what we also know is that
the first dimension of the first matrix and the second dimension of the second
matrix are gonna come together to give us the dimensions of our result. So therefore, our result in this
example is gonna be an 𝑚-by-𝑝 matrix.

So now, we’re looking to solve our
problem. We’re gonna first of all take a
look at our matrices. We’ve got a one by three and the
three by one. Well, to check that they can
actually be multiplied, we check the second dimension of the first matrix and the
first dimension of the second matrix, and they’re the same. So this is okay. Well, then, if we take a look at
the product for 𝐴𝐵, then this is gonna be the remaining dimensions, so it’s gonna
be one by one. And that’s because the product is
𝐴𝐵. So we have our 𝐴 matrix first and
then our 𝐵 matrix. So we’ve got a one-by-one matrix as
the result.

So now, to find out the value of
our one-by-one matrix, what we’re gonna do is take a look at each of our
elements. And then what we do is multiply the
corresponding elements, so the first element in the first matrix by the first
element in the second matrix. So we’re gonna have one multiplied
by eight. And then we do the same for the
second elements. So we have negative two multiplied
by one, and we add this to the first. And then finally, we add the
product of the third corresponding elements. So it’s gonna be negative three
multiplied by negative three, which is gonna give us eight minus two plus nine,
which is gonna give us our final answer for 𝐴𝐵, which is the one-by-one matrix
15. Okay, great! So now let’s move on to find the
product of 𝐵𝐴.

Well, with 𝐵𝐴, we’ve actually got
our matrices the other way round. So we’ve got three-by-one matrix
multiplied by a one-by-three matrix. Well, here, once again, we can see
that the second dimension of the first matrix is the same as the first dimension of
the second matrix. So it means that we can multiply
our matrices. And then the same as before, we can
have a look at the first dimension of our first matrix and the second dimension of
our second matrix. And this is gonna tell us the
dimensions of our result, so it’s gonna be the three-by-three matrix. Okay, great.

So now let’s multiply our matrices
and form our three-by-three matrix. Well, as the same as in the first
part of the question, we’re gonna multiply corresponding elements. But this time, obviously, we need
to make a three-by-three matrix. So the way it’s gonna work, first
of all, is we’re gonna have our first element from our first column, or our only
column, in our first matrix multiplied by the first element in the second
matrix. So it’s gonna be eight multiplied
by one. And that’s the first element of our
answer matrix complete. Now we’re gonna move on to the next
element.

Then the next element is found by
multiplying eight by negative two because what we’ve done is move along to the
second element in our second matrix and then finally multiply our eight by the final
element in our second matrix. So we get eight multiplied by
negative three. And then that’s the first row of
our answer matrix complete. And then, for the second row, we
just move down a row in our original matrix. So now we’re gonna have one, and
then we’re gonna have one multiplied by one. And then we’ve got one multiplied
by a negative two, etcetera. And then we continue this pattern
on to finish our answer matrix. Okay, great! Now all we need to do is work out
each of the individual elements. And when we do that, we’re gonna
get the matrix eight, negative 16, negative 24, one, negative two, negative three,
negative three, six, and nine.