### Video Transcript

Consider the matrices ๐ด equals
zero, negative four, two, negative two, ๐ต equals negative five, six, ๐ฅ, ๐ฆ. If ๐ด๐ต is equal to ๐ต๐ด, what are
the values of ๐ฅ and ๐ฆ?

To answer this question, weโll need
to begin by evaluating ๐ด๐ต and ๐ต๐ด. Remember, these arenโt always the
same due to the fact that matrix multiplication is not commutative. It canโt be performed in any
order. So weโll need to work them out
separately. To find ๐ด๐ต, we find the dot
product of the rows in the first matrix and the columns in the second. Letโs see what that looks like.

To find the first element in the
first row, weโre going to find the dot product of the row zero, negative four and
the column with entries negative five, ๐ฅ. Thatโs zero multiplied by negative
five plus negative four multiplied by ๐ฅ. Zero multiplied by negative five is
zero. So this is simply negative four
๐ฅ. To find the second element of our
first row, we repeat this process, finding the dot product of the first row in the
first matrix and the second column in the second. This time that zero multiplied by
six plus negative four multiplied by ๐ฆ, which is negative four ๐ฆ.

To find the first entry of the
second row, we find the dot product of the elements in the second row and first
column. This time, thatโs two multiplied by
negative five plus negative two multiplied by ๐ฅ, which is negative 10 minus two
๐ฅ. And finally, we find the dot
product of the elements on the second row in the first matrix and the second column
in the second. Thatโs two multiplied by
~~negative six~~ [six] plus negative two multiplied by ๐ฆ, which is 12
minus two ๐ฆ.

Letโs repeat this process for
๐ต๐ด. The first entry is negative five
multiplied by zero plus six multiplied by two, which is 12. The second entry is negative five
multiplied by negative four, which is 20, plus six multiplied by negative two, which
is negative 12. And that simplifies to make
eight. Now, in fact, we donโt need to do
anything more. We can actually solve this
problem. But letโs complete this matrix.

๐ฅ multiplied by zero plus ๐ฆ
multiplied by two is simply two ๐ฆ. And ๐ฅ multiplied by negative four
plus ๐ฆ multiplied by negative two is negative four ๐ฅ minus two ๐ฆ. And weโre told these matrices are
identical. This means each of their individual
elements must be the same. And we can say that negative four
๐ฅ is equal to 12. And negative four ๐ฆ is equal to
eight. And weโll solve these equations to
find ๐ฅ and ๐ฆ, respectively.

To solve this first equation, we
divide by negative four. That gives us that ๐ฅ is equal to
negative three. And similarly, we solve the second
equation by dividing by negative four. And this time, we get ๐ฆ is equal
to negative two. Once we have these, itโs a really
nice way to check our answers. We can substitute the values of ๐ฅ
and ๐ฆ that weโve worked out into the individual elements in our equation.

Substituting ๐ฅ is equal to
negative three into the expression negative 10 minus two ๐ฅ gives us negative
four. And substituting ๐ฆ is equal to
negative two into the expression two ๐ฆ also gives us negative four. Remember, we said the individual
elements had to be equal. So this is a good way to check what
weโve done is correct.

๐ฅ is equal to negative three. And ๐ฆ is equal to negative
two.