### Video Transcript

Take the matrix 𝐴 equals one,
zero, zero, zero, five, zero, zero, zero, two. Then, what we need to do is find 𝐴
squared and then use this to find 𝐴 cubed.

Well, what 𝐴 squared is gonna be
is matrix one, zero, zero, zero, five, zero, zero, zero, two multiplied by the
matrix one, zero, zero, zero, five, zero, zero, zero, two. And we know that they can be
multiplied by each other because the number of columns in matrix one is the same as
the number of rows in matrix two. And that’s because they are both,
in fact, three-by-three matrices. And therefore, our result is also
gonna be a three-by-three matrix.

Now, in order to calculate what the
first element in our matrix is going to be, what we do is we multiply the
corresponding elements from the first row of our first matrix and the first column
of our second matrix and then add them together. So we have one multiplied by one
plus zero multiplied by zero plus zero multiplied by zero.

So then, from the next element
along, what we do is this time look at the second column in our second matrix. So this time, what we’ll have is
one multiplied by zero, zero multiplied by five, and zero multiplied by zero. So then finally, for the last
element in our first row, what we’re gonna have is one multiplied by zero, zero
multiplied by zero, and zero multiplied by two. And we got that because we moved on
now to the third column in our second matrix.

Okay, great. So we now have the process. We use that just to fill in the
rest of the matrix. So when we move to the next row
down, what we do is we just move the row down in the first matrix and then we
restart at the beginning with the first column in the second matrix. So it’s zero multiplied by one plus
five multiplied by zero plus zero multiplied by zero. So then, we just continue our
pattern and use this to complete our matrix. So we’ve done that here. So we’ve now got all of the
calculations in place.

So now, what we need to do is
calculate each of the elements of our matrix 𝐴 squared. Well, our first element is one
cause we got one plus zero plus zero, then zero, then zero again. So that’s our first row
complete. And then for our second row, we’re
gonna get zero, 25, and zero. So then, for our final row, we’re
gonna get zero, zero, four. So great, we found our matrix for
𝐴 squared. And that is one, zero, zero, zero,
25, zero, zero, zero, four.

So now, what we’re gonna do is move
on to 𝐴 cubed. Well, in order to find 𝐴 cubed,
what this is gonna be equal to is the matrix for 𝐴 squared multiplied by the matrix
for 𝐴. So what it’s gonna be is the matrix
one, zero, zero, zero, 25, zero, zero, zero, four multiplied by the matrix one,
zero, zero, zero, five, zero, zero, zero, two. And we do that the same way that we
multiplied two matrices earlier.

So to quickly remind us, we have a
look at the first element. So what we do is we multiply the
corresponding elements from the first row of the first matrix and the first column
of the second matrix and add them together. So what we get is one multiplied by
one add zero multiplied by zero add zero multiplied by zero. So then we use this method to carry
on to complete the rest of our matrix. So we see here that we’ve written
out all the calculations that we need. So our first row of the matrix 𝐴
cubed is gonna be one, zero, zero. The next row will be zero, 125,
zero. So that will give us a final row of
zero, zero, eight.

So great, we could solve the
problem because we can say that if we take the matrix 𝐴, which is one, zero, zero,
zero, five, zero, zero, zero, two, then 𝐴 squared will be one, zero, zero, zero,
25, zero, zero, zero, four. And 𝐴 cubed will be one, zero,
zero, zero, 125, zero, zero, zero, eight.