Question Video: Identifying the Transpose of the Sum of Transpose Matrices Mathematics

Video Transcript

Complete the following. If 𝐴 and 𝐵 are two matrices with the same size, then the transpose of the transpose of 𝐴 plus the transpose of 𝐵 equals what.

Let’s start by defining our matrices 𝐴 and 𝐵. Since every element in the matrices will be constant, we can denote these elements using index notation. So 𝑎 𝑖𝑗 can represent any element in our matrix 𝐴, where 𝑖 will represent the row of the element and 𝑗 will represent the column of the element. Now, we can say that 𝐴 is equal to the matrix of the elements 𝑎 𝑖𝑗. Let’s look at an example of how this can work. Suppose that our matrix 𝐴 is a two-by-two matrix, which is equal to two, zero, five, negative one. Let’s split our matrix up in two rows and columns. From our grid here, we can see that the element in the first row and first column of the matrix is two. Representing this using index notation, this means we can say that 𝑎 one one is equal to two.

Now if we take a look at the first row but the second column, we can see that the element is zero. Therefore, 𝑎 one two is equal to zero. Now, when we look at the second row, we can see in the first column we have five and in the second column we have negative one, giving us that 𝑎 two one is equal to five and 𝑎 two two is equal to negative one. Now we could use this index notation to denote a matrix of any size. So even though we don’t know the size of the matrix 𝐴, we can still say that it is equal to 𝑎 𝑖𝑗. Similarly, we can say that our matrix 𝐵 is equal to the matrix of the elements 𝑏 𝑖𝑗.

Now what we need to find is the transpose of 𝐴 and the transpose of 𝐵. We know that when finding the transpose of a matrix, we take the rows of the original matrix and they become the columns of the transpose matrix. Another way we could think about this is if we take the columns of the original matrix, they can become the rows of the transpose matrix. So essentially what we’re doing is swapping the rows and the columns of the matrix. In index notation, since 𝑖 represents the row and 𝑗 represents the column of the element, all that we need to do is switch 𝑖 and 𝑗 around. So we have that the transpose of 𝐴 is equal to the elements 𝑎 𝑗𝑖 and the transpose of 𝐵 is equal to the elements 𝑏 𝑗𝑖.

Let’s now quickly consider what will happen to the order of our matrices. We’ve been told in the question that 𝐴 and 𝐵 have the same size. So we can say that 𝐴 and 𝐵 are both 𝑚-by-𝑛 matrices, where 𝑚 and 𝑛 are both positive integers. Now, when we take the transpose of a matrix, the order of the matrix will be inverted. Hence, both the transpose of 𝐴 and the transpose of 𝐵 will be of the order 𝑛 by 𝑚. So again, they will be the same size. And this is a good thing since we’re going to want to add these together. And in order to add matrices, we need them to be the same size.

Now when we add matrices together, what we need to do is to add each of the corresponding elements to one another. This means that to find the element in the first row and the first column of the sum of the transposes of 𝐴 and 𝐵, we’ll need to add the elements 𝑎 one one and 𝑏 one one. Similarly, for the element in the first row and second column of the sum, we’ll need to add 𝑎 one two and 𝑏 one two. Here is an expanded version of what the transpose of 𝐴 plus the transpose of 𝐵 should look like. As we can see, we’re adding each of the corresponding elements from the two matrices.

The important thing to note is that the indices of the pairs of elements, which we’re adding together to form the new elements, all match one another within each element. Therefore, we’re able to use the index form of the elements to say that the transpose of 𝐴 plus the transpose of 𝐵 is equal to 𝑎 𝑗𝑖 plus 𝑏 𝑗𝑖.

Now we can simplify this further. Since we have matching indices within each element and each element is still just a constant, we can relabel each of these indices using 𝑐. For example, we can set 𝑎 one one plus 𝑏 one one to be equal to 𝑐 one one and 𝑎 one two plus 𝑏 one two to be equal to 𝑐 one two. And we can also set 𝑎 two one plus 𝑏 two one to be equal to 𝑐 two one. If we do this for every element in our matrix, then we will be able to rewrite 𝑎 𝑗𝑖 plus 𝑏 𝑗𝑖 as 𝑐 𝑗𝑖. And so we arrive at this simplified version of our sum.

Next, we note that the question has asked us to find the transpose of the transpose of 𝐴 plus the transpose of 𝐵. So we now need to take the transpose of our sum. If we remember, when we take the transpose of a matrix where its elements are given in index form, we simply switch the order of the indices. So we can say that the transpose of 𝐴 transpose plus 𝐵 transpose is equal to the matrix of the elements 𝑐 𝑖𝑗. Now we can relate this back to what we started with. Let’s see what happens when we add the matrices 𝐴 and 𝐵. In a similar way to when we added the transposes of 𝐴 and 𝐵, we simply add each corresponding element of 𝐴 to the corresponding element of 𝐵. And we can say that 𝐴 plus 𝐵 is equal to the matrix of the elements 𝑎 𝑖𝑗 plus 𝑏 𝑖𝑗.

Now, from before, we have that 𝑎 𝑗𝑖 plus 𝑏 𝑗𝑖 is equal to 𝑐 𝑗𝑖. And the statement will also hold true if we switch the indices for each of the elements, giving us that 𝑎 𝑖𝑗 plus 𝑏 𝑖𝑗 is equal to 𝑐 𝑖𝑗. Hence, we can rewrite the elements in our sum as 𝑐 𝑖𝑗. We can now see that the matrix of 𝐴 plus 𝐵 matches the matrix of the transpose of the transpose of 𝐴 plus the transpose of 𝐵. This leads us directly to our solution, which is that the transpose of the transpose of 𝐴 plus the transpose of 𝐵 is equal to 𝐴 plus 𝐵.