Given the inverse of 𝐴 is equal to two, five, one, 32, 11, 74, 32, 12, 18, find the inverse of the transpose of 𝐴.
We’ve been given the inverse of matrix 𝐴, and then we’re asked to find the inverse of the transpose of the original matrix. So, what we could do is begin by finding the inverse of the inverse of 𝐴; that will give us the original matrix 𝐴 which we can then transpose and find the inverse of. However, there is a really useful property that we can apply that will save us a lot of time. That is, assuming 𝐴 is an invertible matrix, the transpose of the inverse of 𝐴 is equal to the inverse of the transpose of 𝐴. And this means that we can find the inverse of the transpose of matrix 𝐴 by simply transposing the inverse 𝐴.
So, how do we transpose a matrix? The elements on the main diagonal remain unchanged and then we switch the elements across this diagonal. This corresponds essentially to switching the rows and the columns. So, let’s take the elements in our first row, and we’re going to put them in our first column, as shown. Next, we take the elements in our second row and we put them in our second column. That gives us a completed second column of our transpose. We do this one more time, taking the elements in our third row of our first matrix and transposing them into our third column. And when we do, we have the inverse of the transpose of 𝐴.
The transpose of the inverse, which is also the inverse of the transpose, is the three-by-three matrix with elements two, 32, 32, five, 11, 12, one, 74, 18.