Consider the matrices 𝐴 equals
zero, negative four, two, negative two, 𝐵 equals negative five, six, 𝑥, 𝑦. If 𝐴𝐵 is equal to 𝐵𝐴, what are
the values of 𝑥 and 𝑦?
To answer this question, we’ll need
to begin by evaluating 𝐴𝐵 and 𝐵𝐴. Remember, these aren’t always the
same due to the fact that matrix multiplication is not commutative. It can’t be performed in any
order. So we’ll need to work them out
separately. To find 𝐴𝐵, we find the dot
product of the rows in the first matrix and the columns in the second. Let’s see what that looks like.
To find the first element in the
first row, we’re going to find the dot product of the row zero, negative four and
the column with entries negative five, 𝑥. That’s zero multiplied by negative
five plus negative four multiplied by 𝑥. Zero multiplied by negative five is
zero. So this is simply negative four
𝑥. 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 𝑦, which is negative four 𝑦.
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, that’s two multiplied by
negative five plus negative two multiplied by 𝑥, which is negative 10 minus two
𝑥. 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. That’s two multiplied by
six plus negative two multiplied by 𝑦, which is 12
minus two 𝑦.
Let’s repeat this process for
𝐵𝐴. 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 don’t need to do
anything more. We can actually solve this
problem. But let’s complete this matrix.
𝑥 multiplied by zero plus 𝑦
multiplied by two is simply two 𝑦. And 𝑥 multiplied by negative four
plus 𝑦 multiplied by negative two is negative four 𝑥 minus two 𝑦. And we’re told these matrices are
identical. This means each of their individual
elements must be the same. And we can say that negative four
𝑥 is equal to 12. And negative four 𝑦 is equal to
eight. And we’ll solve these equations to
find 𝑥 and 𝑦, respectively.
To solve this first equation, we
divide by negative four. That gives us that 𝑥 is equal to
negative three. And similarly, we solve the second
equation by dividing by negative four. And this time, we get 𝑦 is equal
to negative two. Once we have these, it’s a really
nice way to check our answers. We can substitute the values of 𝑥
and 𝑦 that we’ve worked out into the individual elements in our equation.
Substituting 𝑥 is equal to
negative three into the expression negative 10 minus two 𝑥 gives us negative
four. And substituting 𝑦 is equal to
negative two into the expression two 𝑦 also gives us negative four. Remember, we said the individual
elements had to be equal. So this is a good way to check what
we’ve done is correct.
𝑥 is equal to negative three. And 𝑦 is equal to negative