Video Transcript
Consider the matrices matrix 𝐴 is
the three-by-one matrix one, one, one. The transpose of matrix 𝐴 is the
one-by-three matrix one, one, one. Matrix 𝐵 is the one-by-three
matrix 𝑎, 𝑏, 𝑐, and the transpose of matrix 𝐵 is the three-by-one matrix 𝑎, 𝑏,
𝑐. Then, find 𝐴𝐵 and find the
transpose of 𝐵 multiplied by the transpose of 𝐴.
Well, now, if we consider 𝐴𝐵
first, then what we can see straightaway is that we can multiply the matrix 𝐴 by
the matrix 𝐵. And we can do that because the
number of columns in matrix 𝐴 is equal to the number of rows in matrix 𝐵. And what we also know is that our
result matrix is going to be a three-by-three matrix. And that’s because there are three
rows in matrix 𝐴 and three columns in matrix 𝐵. So therefore, 𝐴𝐵 is going to be
equal to the three-by-one matrix one, one, one multiplied by the one-by-three matrix
𝑎, 𝑏, 𝑐.
So now, to find the first row of
our answer matrix, we’re going to multiply the first element of the first row in our
first matrix by each of the elements in our other matrix. So we’re gonna have one multiplied
by 𝑎, one multiplied by 𝑏, and one multiplied by 𝑐. And then, due to the nature of the
matrices that we’re looking at, our following two rows are going to be exactly the
same. So we’re going to have one
multiplied by 𝑎, one multiplied by 𝑏, and one multiplied by 𝑐 for each of those
two rows. And so this is gonna give us the
three-by-three answer matrix 𝑎, 𝑏, 𝑐, 𝑎, 𝑏, 𝑐, 𝑎, 𝑏, 𝑐.
So, great, we’ve now solved the
first part of the question. What we’re going to do is move on
to the second part. And what we’re gonna do here is
multiply the transpose of matrix 𝐵 by the transpose of matrix 𝐴. So once again, we’re going to have
a three-by-one matrix multiplied by a one-by-three matrix. And we know that they are going to
multiply. And that’s because, again, we’ve
got the same number of columns in the first matrix as the number of rows in the
second matrix. And once again, we know that our
answer matrix is going to be a three-by-three matrix. And that’s because the number of
rows in our first matrix is three. And that’s the first matrix, which
is the transpose of matrix 𝐵. And the number of columns in our
second matrix, which is the transpose of 𝐴, is also three.
So what we can say is that the
transpose of matrix 𝐵 multiplied by the transpose of matrix 𝐴 is going to be equal
to the three-by-one matrix 𝑎, 𝑏, 𝑐 multiplied by the one-by-three matrix one,
one, one. So this time what we’re going to do
once again is multiply the first element from the first row in the first matrix by
each of the elements in our second matrix. So we’re gonna get 𝑎 multiplied by
one, 𝑎 multiplied by one, and 𝑎 multiplied by one. And this is the first row of our
answer matrix. Then our second row is going to be
𝑏 multiplied by one, 𝑏 multiplied by one, 𝑏 multiplied by one. And then the final row is going to
be 𝑐 multiplied by one, 𝑐 multiplied by one, 𝑐 multiplied by one. And this is going to give us our
three-by-three answer matrix 𝑎, 𝑎, 𝑎, 𝑏, 𝑏, 𝑏, 𝑐, 𝑐, 𝑐.
So we now solved both parts of the
question. However, there’s something
interesting that we might notice between the two answers. If we look at the answer to the
first part, we’re gonna have the three-by-three matrix 𝑎, 𝑏, 𝑐, 𝑎, 𝑏, 𝑐, 𝑎,
𝑏, 𝑐. Well, then, if we take the
transpose of this matrix. And what the transpose means is
that we switch the corresponding rows and columns. And for a square matrix, this is
the same as swapping elements over the main diagonal. And if we do this, we’ll see that
we’ll get the three-by-three matrix 𝑎, 𝑎, 𝑎, 𝑏, 𝑏, 𝑏, 𝑐, 𝑐, 𝑐, which is in
fact the answer we got when we multiplied together the transpose of matrix 𝐵 by the
transpose of matrix 𝐴.
What we have in fact shown here is
an example of the matrix property that states that the transpose of 𝐴𝐵 is equal to
the transpose of 𝐵 multiplied by the transpose of 𝐴.
