Video Transcript
Consider the matrices 𝐴 is a matrix negative four, two, negative six, negative six, 𝐵 is negative five, negative one, one, zero. Find the transpose of 𝐴 multiplied by matrix 𝐵 and matrix 𝐴 multiplied by the transpose of 𝐵.
Now the first thing we need to do in this question is find the transposes of both of our matrices. So let’s start with the transpose of the matrix 𝐴. So the first thing we notice when working out the transpose of a matrix is that the diagonal, that is, from the top left to the bottom right, remains the same. So therefore, we can see that the top-left element and bottom-right elements of our transpose are going to be negative four and negative six, respectively.
And then what we do is swap the remaining elements across our diagonal. So when we do this, what we’re gonna get is the matrix negative four, negative six, two, and negative six as the transpose of 𝐴. So what we can see is the only thing that’s happened here is the two and the negative six have swapped places. And it’s as simple as that when we’re looking at a two-by-two matrix.
Okay, so now let’s move on to the transpose of matrix 𝐵. Well, first of all, once again, we can see that the diagonals are gonna remain the same. So our top-left and bottom-right elements are going to be negative five and zero, respectively. And then, once again, what we do is we swap the remaining elements over our diagonal. So we’re gonna end up with the transpose being the matrix negative five, one, negative one, and zero. Great, so now we’ve got the transpose of matrix 𝐴 and the transpose of matrix 𝐵, we can actually get on and solve the problem.
So first of all, what we’re gonna do is multiply the transpose of matrix 𝐴 by the matrix 𝐵. So we’re gonna have the matrix negative four, negative six, two, negative six multiplied by the matrix negative five, negative one, one, zero.
So before we actually multiply our matrices, let’s remind ourselves a little bit about multiplying matrices. So first of all, to be able to multiply two matrices together, the second dimension of the first matrix must be the same as the first dimension of the second matrix. So in our problem, that’s no problem because we’ve got two two-by-two matrices, so they’re going to be the same. Then what we also know is that the remaining dimensions give us our dimensions or order of our result. So we can see here in the example, if we have an 𝑚-by-𝑛 matrix multiplied by an 𝑛-by-𝑝 matrix, then the result is gonna be an 𝑚-by-𝑝 matrix.
So then if we apply that to our matrices, what we’ve got is two two-by-two matrices. So therefore, our result is gonna be a two-by-two matrix, which is what we would expect. So now what we’re gonna do is look at how we multiply the matrices and work out what the individual elements are. Now to work out what the individual elements actually are, what we’re going to do is multiply some corresponding elements.
So to work out the first element, what we’re gonna do is multiply the first element in the first row of the first matrix by the first element in the first column of the second matrix, so that’s gonna be negative four multiplied by negative five. Then we’re gonna add to it the second element in the first row of the first matrix multiplied by the second element in the first column of the second matrix, so negative six multiplied by one.
So then, for the next element, what we do is move along. So we’re gonna have negative four multiplied by negative one plus negative six multiplied by zero. So now what we’re gonna do is move to the second row of the first matrix. So we’re gonna have two multiplied by negative five plus negative six multiplied by one. And then for the final element, what we’re gonna have is two multiplied by negative one add negative six multiplied by zero. So then all we need to do is calculate all of the individual elements. And when we do, we’re gonna get the matrix 14, four, negative 16, negative two.
Well, then that’s the first part of the question answered because we’ve got the transpose of matrix 𝐴 multiplied by matrix 𝐵. So now let’s move on to the next part of the question, which is the matrix 𝐴 multiplied by the transpose of the matrix 𝐵. So in order to do this, we’re gonna clear some space. So therefore, what we’re gonna have is the matrix negative four, two, negative six, negative six multiplied by the matrix negative five, one, negative one, zero. And that’s cause that’s the transpose of the matrix 𝐵.
So then, using the same method as before, our first element is gonna be negative four multiplied by negative five plus two multiplied by negative one. And then all we need to do next is continue this pattern to work out the calculations for each of our other elements, which you can see here. And when we calculate each of these, we’re gonna be left with the matrix 18, negative four, 36, negative six.
And therefore, we’ve solved both parts of the problem because we can say that the transpose of matrix 𝐴 multiplied by the matrix 𝐵 and the matrix 𝐴 multiplied by the transpose of matrix 𝐵 are the matrices 14, four, negative 16, negative two and 18, negative four, 36, and negative six, respectively.
