True or False: Entries on the main diagonal of matrix A Q. If $A = [a_{ij}]$ is an $m \times n$ matrix which satisfies $A^T = -A$, then the entries on the main diagonal of $A$ are all equal to $0$.
I don't see how $A^T = -A$ can be true for a $m \times n$ matrix. Also, two matrices are only equal if they have the same size (dimensions) and have the same entries. If A is a m x n matrix, then so will -A. However, AT would be a n x m matrix, so $A^T \neq -A$. 
So I'm not sure on how to proceed with this question. Any help would be appreciated.

Firstly, for this question to make any sense (that is, that the statement $A^T=-A$ is meaningful) we must have $m=n$. Now let's examine the diagonal entries, $a_{ii}$. The $i, j$-th element of $A^t$ is the $j, i$-th element of $A$. In particular (for $i=j$) the diagonal entries of $A^t$ are the same as the diagonal entries of $A$. So for any $1\le i\le n$, we have $a_{ii}=-a_{ii}$, since $A^T=-A$, which implies $a_{ii}=0$, as desired.
