### Video Transcript

Consider the matrices π΄ equals
zero, negative four, two, negative two, π΅ equals negative five, six, π₯, π¦. If π΄π΅ is equal to π΅π΄, what are
the values of π₯ and π¦?

To answer this question, weβll need
to begin by evaluating π΄π΅ and π΅π΄. Remember, these arenβt always the
same due to the fact that matrix multiplication is not commutative. It canβt be performed in any
order. So weβll need to work them out
separately. To find π΄π΅, we find the dot
product of the rows in the first matrix and the columns in the second. Letβs see what that looks like.

To find the first element in the
first row, weβre going to find the dot product of the row zero, negative four and
the column with entries negative five, π₯. Thatβs zero multiplied by negative
five plus negative four multiplied by π₯. Zero multiplied by negative five is
zero. So this is simply negative four
π₯. 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 π¦, which is negative four π¦.

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, thatβs two multiplied by
negative five plus negative two multiplied by π₯, which is negative 10 minus two
π₯. 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. Thatβs two multiplied by
~~negative six~~ [six] plus negative two multiplied by π¦, which is 12
minus two π¦.

Letβs repeat this process for
π΅π΄. 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 donβt need to do
anything more. We can actually solve this
problem. But letβs complete this matrix.

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

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

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

π₯ is equal to negative three. And π¦ is equal to negative
two.