Given that matrix 𝐴 is negative two, negative nine, eight, seven, find the inverse of the transpose of matrix 𝐴.
When trying to solve this problem, we can actually notice two bits of notation here. We’ve got 𝐴 to capital 𝑇, and then that is all in our parenthesis to negative one. So what I actually have is two things. We have the transpose and the inverse. And we’re gonna actually have to find both of those to solve this problem. We can start first of all with the transpose.
So when we’re looking to actually transposing a matrix, what we can actually say is for each element of 𝑎𝑥𝑦 where 𝑥 is the 𝑥th row and 𝑦 is the 𝑦th column. In our matrix 𝐴, the matrix 𝐴𝑇, so our transposed matrix, has an equal element at 𝑎𝑦𝑥, so this time at the 𝑦th row and the 𝑥th column. So what does that actually mean in practice?
Well in practice, what it actually means is that we’re going to turn the rows into their corresponding columns. For example, we’re gonna to turn the first row of our matrix into the first column of its transpose. So if we take a look at our matrix 𝐴, so what we’ve actually got is that matrix 𝐴 is negative two, negative nine, eight, seven. And we’ve got our first row, which is negative two, negative nine. And we’ve actually then turned that into our first column of our transpose matrix. So it’s gonna to be negative two, negative nine.
And now we’ve done the same thing with our second row. So our second row — eight, seven — has become our second column of our transposed matrix, again eight seven reading downwards. So therefore we could say that 𝐴𝑇 is equal to negative two, eight, negative nine, seven. So now we’re gonna have a look at the inverse of a matrix cause we wanna see how do we find that. Well first of all, a bit like when you multiply a number by its inverse you get one, with matrix if you multiply a matrix by its inverse, you actually get 𝐼 where 𝐼 is actually called the identity matrix which can be represented by the matrix one, zero, zero, one.
Well, okay! So we know that that’s the case, but how we’re actually gonna find the inverse of our matrix. Well, actually what we’ve got is a two-by- two matrix. And there’s a special formula to help us find the inverse of a two-by-two matrix. And what this formula actually tells us is that if we have a matrix in the form 𝑎, 𝑏, 𝑐, 𝑑, then the universe of this matrix is equal to one over 𝑎𝑑 minus 𝑏𝑐, which is the determinant of that matrix, multiplied by the matrix 𝑑, negative 𝑏, negative 𝑐, 𝑎 where we’ve swapped our 𝑎 and 𝑑 values and our 𝑏 and 𝑐 elements are now negative.
Okay, so now we’ve got this formula. Let’s use it to actually find the inverse of the transposition of matrix 𝐴. Therefore, as our transpose matrix 𝐴 is going to be negative two, eight, negative nine, seven, then the inverse of that transposition is gonna be equal to one over and then it’s 𝑎 multiplied by 𝑑, which is negative two multiplied by seven, and then 𝑏 multiplied by 𝑐, which will be eight multiplied by negative nine. Okay, and then this is all multiplied by the matrix, with our first element as seven because that’s our 𝑑. Our second element is going to be a negative eight because it’s the negative of the 𝑏 element, so negative eight.
And then our third element, the bottom left element, is gonna be nine because our 𝑐 element in the transpose matrix is negative nine. And this one is negative 𝑐, so negative negative nine gives us positive nine. And then our final element is negative two. Okay, great! So let’s calculate. So we get that it’s all equal to one over 58 multiplied by the matrix seven, negative eight, nine, negative two, which gives us the matrix seven over 58 negative eight, over 58, nine over 58, negative two over 58. And therefore, if we simplify our fractions fully, we can say that given that the matrix 𝐴 is equal to negative two, negative nine, eight, seven, then the inverse of the transposition of matrix 𝐴 is gonna be equal to seven over 58, negative four over 29, nine over 58, and negative one over 29.