Video Transcript
Given that 𝐴 inverse equals two,
negative five, negative eight, negative nine; 𝐴 multiplied by 𝑥, 𝑦 equals
negative two, eight, what is the value of 𝑦?
In this matrix equation, we don’t
know the matrix 𝐴 and we don’t know the values for 𝑥 and 𝑦, but we have been given
𝐴 inverse. So let’s think about how we could
use 𝐴 inverse to help us. Ideally, if we could get 𝑥, 𝑦 on
its own, we might be able to solve for 𝑦 from there. In fact, this is where the inverse
of 𝐴 is going to come in handy. If we start by multiplying both
sides of the matrix equation by 𝐴 inverse, we end up with 𝐴 inverse multiplied by
𝐴. And you’ll recognize this as a
general result that we have for matrices.
The inverse of a matrix multiplied
by the matrix itself always gives us the identity matrix. So instead of 𝐴 inverse multiplied
by 𝐴, we just have 𝐼. And as another general result we
have for matrices, when we take the identity matrix and multiply it by the matrix
𝐴, we just get the matrix 𝐴. So we can rewrite 𝐼 multiplied by
𝑥, 𝑦 as just 𝑥, 𝑦. And now this is looking like
something that’s a little more simple to solve. We were given the matrix 𝐴 inverse
in the question. Now from here, let’s just simplify
what we have on the right-hand side of this matrix equation. We can do this by multiplying the
matrix two, negative five, negative eight, negative nine with the matrix negative
two, eight.
We know that this matrix
multiplication must be possible because the matrix on the left is a two-by-two
matrix and the matrix on the right is a two-by-one matrix. And because these two numbers are
the same, this multiplication is going to be possible and will result in a matrix of
order two by one. So let’s go ahead with the matrix
multiplication.
First, we have two multiplied by
negative two, which is negative four. And then we add negative five
multiplied by eight. That gives us negative 40. And on the bottom, we have negative
eight multiplied by negative two. That gives us 16. And then we add negative nine
multiplied by eight. That gives us negative 72. And finally, let’s simplify both
the entries in this matrix. That gives us the matrix negative
44, negative 56. The original question asked us for
the value of 𝑦. So we can see that 𝑦 is negative
56. So that’s our answer, 𝑦 equals
negative 56.
