Here we're asked to find
the product of two matrices.
The first step is to see if the product is possible.
In order to multiply two matrices, the number of columns
in the first matrix must be equal to the number of rows
in the second matrix.
So for this example, we have a two-by-two matrix,
times a two-by-two matrix.
So to determine if the product is possible,
we want to focus on these two values here.
Again, the number of columns in the first matrix
must equal the number of rows in the second matrix.
Which, in this case, is true.
And the remaining values, or the number of rows in
the first matrix, and the number of columns
in the second matrix indicate the dimensions of the product.
So the product of these two matrices
will be another two-by-two matrix.
Let's go ahead and set this up below.
Again, we know this product is possible,
and the product will be a two-by-two matrix.
And now to find the elements in this product,
we always multiply rows from the first matrix
and columns from the second matrix.
So to determine the element in row one, column one,
or A sub one, comma, one,
we multiply row one from the first matrix
and column one from the second matrix.
So again, we'll have this row times this column.
To multiply a row and a column,
we find the sum of the products of the first elements,
second elements, third elements, and so on.
Or in this case, we find the sum of the product
of the first elements and the second elements.
So A sub one, comma, one, is equal
to negative four times negative two
plus negative three times four.
So here we have positive eight plus negative 12,
which equals negative four.
So the element in row one, column one of the product
is negative four.
And now let's find the element in row one, column two,
which is A sub one, comma, two.
Which means we'll multiply row one from the first matrix,
and column two from the second matrix.
So row one from the first matrix,
column two from the second matrix.
So the result would be negative four times three,
plus negative three times one,
which is equal to negative 12 plus negative three,
which equals negative 15.
And now let's find the element in row two, column one.
For A sub two, comma, one, we'll multiply row two
from the first matrix and column one from the second matrix.
Again, row two from the first matrix,
and column one from the second matrix.
So we'll have negative two times negative two,
plus negative two times four,
which is equal to positive four plus negative eight,
which equals negative four.
And the last element is in row two, column two.
So to find the element in row two, column two,
we multiply row two from the first matrix,
and column two from the second matrix.
We're going to be multiplying row two from the first matrix,
and column two from the second matrix.
So A sub two, comma, two is equal to
negative two times three, plus negative two times one,
which equals negative six plus negative two,
which equals negative eight.
This is the product of the two matrices.
I hope you found this helpful.