Question Video: Solving Equations of Matrices Using Their Inverses Mathematics

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.