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.
 A: As said in the comments, we need $m = n$ for this question to make sense. Because if not, then $A^T$ would have dimension $n \space x \space m$ which would equal the matrix $-A$ which would have dimension $m \space x \space n$. This is a contradiction since matrices of different sizes cannot equal each other.
Now, if we do assume the correct dimensions, note that when you transpose a matrix, the entries along the diagonal stay the same. So by our assumption, we have that for a diagonal entry $a_{ii}$, it must equal $-a_{ii}$. The only number to have such a property is $0$.
A: First, given a matrix $A$ of size $m$ by $n$, its transpose $A^T$ has size $n$ by $m$. So $A^T=-A$ implies $n=m$.
Further, $(A)_{ii}=(A^T)_{ii}=(-A)_{ii}=-A_{ii}$, for any $i$. I have used the definition of the transpose for the first equality (the coefficients on the diagonal don't change when you transpose a matrix), and your hypothesis on the second. 
So you have that $A_{ii}=-A_{ii}$, which can only happen if $A_{ii}=0$. Meaning that your diagonal coefficents are all $0$.
A: 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.
