Complete the following: 𝐴 plus negative 𝐴 transpose is equal to blank.
The first thing that we notice is that we have a plus negative 𝐴. Now, even though we’re applying these operations to matrices, the plus and the minus still interact with each other in the same way as if these were just numbers. So we can say that 𝐴 plus negative 𝐴 transpose is equal to 𝐴 minus 𝐴 transpose. Next, we can deal with the 𝐴 minus 𝐴. We’re subtracting a matrix from itself. Now, no matter what the order of 𝐴 or the values of the elements of 𝐴, when we subtract this matrix 𝐴 from itself, we will always get the zero matrix.
For example, consider a two-by-two matrix where the entries are 𝑎, 𝑏, 𝑐, and 𝑑. And then we subtract this matrix from itself. Then we can simplify this by subtracting each element from itself, which we can see will leave us with a zero matrix. So going back to our question, we have that 𝐴 minus 𝐴 is just the zero matrix. Since we do not know the order of 𝐴, we do not know the order of the zero matrix. However, this does not really matter, since there is a zero matrix of every dimension.
Now all that remains to do is to find the transpose of this zero matrix. When taking the transpose of a matrix, all the rows of the old matrix become the columns of the new matrix. Since all the entries in the zero matrix are zero, this means that even when we switch the rows and columns of the zero matrix, it will still contain only zeros. The only thing that will change is the dimension of this matrix. However, as discussed before, there is a zero matrix of each dimension. So we can say that the transpose of the zero matrix is just another zero matrix. Here we have reached our solution, which is that the transpose of 𝐴 plus negative 𝐴 is equal to the zero matrix.