Question Video: Finding an Element of the Transpose of a Given Matrix Mathematics

Video Transcript

Consider the matrices 𝐴, which is equal to negative two, 10, seven, one, nine, zero, and 𝐵, which is equal to 𝐴 transpose. Determine 𝑎 subscript one two and 𝑏 subscript one one.

Now, just to remind you about this notation, when we see a matrix and then a superscript 𝑇 as we have here, that means we’re being asked to find the transpose of this matrix. That’s the matrix we get if we swap the rows and columns around. We’ll look at how to do this in a moment because we also need to remind ourselves what is meant by the notation 𝑎 and then a subscript one two and also the notation 𝑏 and then a subscript one one. We recall that a lowercase 𝑎 and then the subscript 𝑚𝑛 defines a particular element of the matrix uppercase 𝐴. It’s the element in row 𝑚 and column 𝑛. Remember, we always list the rows first and the columns second.

So 𝑎 and then a subscript one two means we’re being asked to give the element, which is in the first row and second column of the matrix 𝐴. And 𝑏 and then subscript one one means we’re being asked to give the element in the first row and first column of the matrix 𝐵. Let’s consider 𝑎 subscript one two first of all. We’re looking in the first row and the second column. We can see that the element here is the number 10. So this is 𝑎 subscript one two. Now, let’s consider how we can find 𝑏 subscript one one. And there are two different ways that we could do this. Firstly, as 𝐵 is equal to 𝐴 transpose, we could actually find this matrix. So we could find the transpose of our matrix negative two, 10, seven, one, nine, zero.

Now, as 𝐴 is a matrix with two rows and three columns, when we swap the rows and columns around, its transpose, the matrix 𝐵, will be a matrix with three rows and two columns. The first row of the matrix 𝐴 will form the first column of its transpose. So we have negative two, 10, seven as the first column of matrix 𝐵. The second row of matrix 𝐴 will form the second column of its transpose. So the second column of matrix 𝐵 is one, nine, zero. We’ve found the matrix 𝐵 then. It is equal to negative two, one, 10, nine, seven, zero. To find 𝑏 subscript one one, remember, that’s the element in the first row and first column of matrix 𝐵. So looking at the matrix we found, we can see that this is equal to negative two.

So we’ve answered the question. But is there actually an easier way to answer the second part without needing to find the matrix 𝐵 explicitly? Well, we could consider what element 𝑏 one one would be equal to in the matrix 𝐴. Remember, we swapped the position of the rows and columns around. So in order to work out where this element would have been in matrix 𝐴, we can just swap the two subscript numbers around. In this case, though, they’re both the same. So swapping one and one still gives one one. And we find that 𝑏 subscript one one is the same as 𝑎 subscript one one. And returning to our matrix 𝐴, we can see that this is indeed the case.

Suppose you’ve been asked for an element such as 𝑏 three two, however, which, from our matrix 𝐵, we can see is zero. It’s the element in the third row and second column. We could have found this from the matrix 𝐴 by swapping the two numbers around to give 𝑎 two three and then looking for the element that was in the second row and third column of matrix 𝐴, which we see, once again, is equal to zero. So there was no real need to write out the matrix 𝐵, although it was good practice. We found that 𝑎 subscript one two is equal to 10 and 𝑏 subscript one one is equal to negative two.