We're asked to find the product
of the two matrices.
So 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 notice how this first matrix
is a two by three matrix
because there are two rows and three columns.
The second matrix is a three by four matrix
because there are three rows and four columns.
So if the product is possible,
these two values here must be equal.
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.
So the product is possible.
And these two values here,
or the number of rows in the first matrix
and the number of columns in the second matrix,
give us the dimensions of the product.
So this product will be a two by four matrix.
Again, this product will be a two by four matrix,
meaning we'll have two rows and four columns.
And now to determine the element in this product,
we will always multiply rows in the first matrix
and columns in the second matrix.
So let's go ahead and find the element
in row one, column one,
which should be a sub one comma one.
To do this,
we multiply row one in the first matrix
and column one in the second matrix.
To multiply rows and columns,
we find the sum of the products
of the first elements,
second elements,
third elements and so on.
Or in this case, because the rows and columns
contain three elements,
we'll be summing three products.
So a sub one comma one
is equal to negative two times zero,
plus three times one,
plus negative four times negative four.
So simplifying, we have zero
plus three
plus 16, which equals 19.
So the element in row one, column one
of the product is 19.
And now let's find the element
in row one, column two,
or a sub one comma two.
So we'll multiply row one
from the first matrix,
and now column two from the second matrix.
So we'll have negative two times negative three,
plus three times three,
plus negative four times four.
So simplifying, we have six plus nine,
plus negative 16 or just minus 16,
which is equal to negative one.
So the element in row one, column two is negative one.
And now let's find the element
in row one, column three
or a sub one comma three.
So now we multiply row one
from the first matrix
and column three from the second matrix.
So we'll have negative two times four,
plus three times zero,
plus negative four times three.
Simplifying, we have negative eight plus zero.
So it'd be plus negative 12 or minus 12,
which equals negative 20.
And to finish off this first row,
we need to find the element in row one, column four,
or a sub one comma four.
So to find this element,
we multiply row one from the first matrix
and column four from the second matrix.
So we'd have negative two times negative one,
plus three times negative five,
plus negative four times negative two.
Simplifying, we have two
plus negative 15 or minus 15,
and this would be plus eight,
which equals negative five.
Now let's find the elements in row two.
So let's first find the element
in row two, column one,
or a sub one comma two.
To find the element in row two, column one,
we now multiply row two
from the first matrix
and column one from the second matrix.
So we'd have four times zero,
plus zero times one,
plus negative one times negative four.
We just have zero plus zero plus four,
which equals four.
Next, we have the element in row two, column two.
So we multiply row two from the first matrix
and column two from the second matrix.
So we'd have four times negative three,
plus zero times three,
plus negative one
times four.
So we have negative 12
plus zero
plus negative four or minus four,
which equals negative 16.
Next, we have the element in row two, column three.
So you multiply row two from the first matrix
and column three from the second matrix
so we have four times four,
plus zero times zero,
plus negative one times three.
So we have 16 plus zero.
This would be minus three,
which equals positive 13.
And the last element we need to find
is the element in row two, column four.
So we multiply row two from the first matrix
and column four from the second matrix,
which would be four times negative one,
plus zero times negative five,
plus negative one times negative two.
Simplifying, we have negative four plus zero.
This would be plus two,
which equals negative two.
So this is the product
of the two matrices.
I hope you found this helpful.