Video: Solving a Matrix Equation by Finding the Inverse

Given 𝐴(10, 2, and 10, 1) = (1, 0, and 0, 1), find the matrix 𝐴.

02:43

Video Transcript

Given 𝐴 multiplied by 10, two, 10, one equals one, zero, zero, one, find the matrix 𝐴.

Let’s have a look at what we’ve actually been given here. We’ve been given a matrix equation. We know that the product of 𝐴 with our matrix 10, two, 10, one is one, zero, zero, one. That’s the identity matrix. We can solve this equation a little like we would solve a normal linear equation. We perform inverse operations. Here, since we’re trying to solve for 𝐴, we’re going to multiply both sides of our equation by the inverse of 10, two, 10, one. Now, I’ve called 10, two, 10, one 𝐵. So we’re going to multiply both sides of this equation by the inverse of 𝐵. However, 𝐵 multiplied by the inverse of 𝐵 is the identity matrix.

So on the left-hand side, we’ll simply be left with 𝐴. And the identity matrix multiplied by the inverse of 𝐵 is just the inverse of 𝐵. So we can see that, for our equation, 𝐴 is equal to the inverse of 𝐵. So we just need to find that inverse. 𝐵 is a two-by-two matrix. Let’s call it in its general form 𝑎, 𝑏, 𝑐, 𝑑. Its inverse is one over the determinant of 𝐵 multiplied by 𝑑, negative 𝑏, negative 𝑐, 𝑎. Essentially, we swap the elements on the top left and bottom right. And we change the sign of the elements on the top right and bottom left.

But what about the determinant of 𝐵? The determinant of 𝐵 is the product of 𝑎 and 𝑑 minus the product of 𝑏 and 𝑐. That’s the product of the elements on the top left and bottom right minus the product of the elements on the top right and bottom left. So we’ll begin then by finding the determinant of the matrix 10, two, 10, one. We start by multiplying the elements in the top left and the bottom right. That’s 10 times one. And we subtract the product of the elements in the top right and bottom left. That’s two times 10. 10 minus 20 is negative 10.

So the determinant of the matrix that we’ve called 𝐵 is negative 10. So the first part of the inverse of 𝐵 is one over the determinant of 𝐵. It’s one over negative 10 which is negative one-tenth. We’re then going to switch the elements in the top left and bottom right. And we’re going to change the signs of the elements in the top right and bottom left.

So the inverse of 𝐵 is negative one-tenth multiplied by one, negative two, negative 10, 10. All that’s left is to multiply each element inside our matrix by negative one-tenth. And when we do, we see that the inverse of 𝐵 is negative one-tenth, one-fifth, one, and negative one. And since we said earlier that 𝐴 is going to be equal to the inverse of 𝐵, then 𝐴 is this matrix. It’s negative a tenth, one-fifth, one, negative one.

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