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# Multiplying matrices

## Video transcript

We're given two matrices over
here, matrix E and matrix D. And they ask us,
what is ED, which is another way of saying
what is the product of matrix E and matrix D? Just so I remember
what I'm doing, let me copy and paste this. And then I'm going to get
out my little scratch pad. So let me paste that over here. So we have all the
information we needed. And so let's try
to work this out. So matrix E times matrix D,
which is equal to-- matrix E is all of this business. So it is 0, 3, 5, 5, 5, 2 times
matrix D, which is all of this. So we're going to multiply
it times 3, 3, 4, 4, negative 2, negative 2. Now the first thing
that we have to check is whether this is
even a valid operation. Now the matrix multiplication
is a human-defined operation that just happens-- in
fact all operations are-- that happen to have
neat properties. Now the way that us humans have
defined matrix multiplication, it only works when we're
multiplying our two matrices. So this right over here has
two rows and three columns. So it's a 2 by 3 matrix. And this has three rows and
two columns, it's 3 by 2. This only works-- we
could only multiply this matrix times this matrix,
if the number of columns on this matrix is equal to the
number of rows on this matrix. And in this situation it is, so
I can actually multiply them. If these two numbers weren't
equal, if the number of columns here were not equal to the
number of rows here, then this would not be a valid
operation, at least the way that we have defined
matrix multiplication. The other thing you
always have to remember is that E times D is not always
the same thing as D times E. Order matters when you're
multiplying matrices. It doesn't matter if you're
multiplying regular numbers, but it matters for matrices. But let's actually
work this out. So what we're going to get
is actually going to be a 2 by 2 matrix. But I'm going to
create some space here because we're going to have
to do some computation. So this is going to
be equal to-- I'm going to make a huge
2 by 2 matrix here. So the way we get
the top left entry, the top left entry
is essentially going to be this row
times this product. If you view them
each as vectors, and you have some familiarity
with the dot product, we're essentially
going to take the dot product of that and that. And if you have no idea what
that is, I'm about to show you. This entry is going to be
0 times 3, plus 3 times 3, plus 5 times 4. So that is the top left entry. And I already see that I'm going
to run out of space here, so let me move this
over to the right some space so I have
some breathing room. Now we can do the
top right entry. This was the top left, now
we're going to do the top right. So the top right
entry is going to be this row times this column. Notice the entry
is getting the row from the first matrix and the
column from the second one. That's kind of
determining its position. So, once again, is
going to be 0 times 4, plus 3 times negative 2,
plus 5 times negative 2. And we keep going. The bottom left entry is going
to be this row, the second row here times the
first column here. So it's 5 times 3, plus 5
times 3, plus 2 times 4. And we're almost done. We just need to multiply or
take the dot product of this row with this column
right over here. So it's going to be 5 times
4, plus 5 times negative 2, plus 2 times negative 2. And so this is going
to be equal to, and we could just
evaluate this now. Let's see, 0 times 3 is 0. This is 9 plus 20. This is 29. This all simplified to 29. All of this is 0. This is negative 6. And then this is minus 10. So this all simplifies
to negative 16. This right over here, it's 15
plus 15, which is 30 plus 8. So this is 38. And then finally, this
is 20 minus 10 minus 4. So this is going to be 6. So this is all going
to simplify to 6. So this all became 29,
negative 16, 38, and 6. Let's check our answer. And we got it right.