Ada Matrix

Ada has two matrices $A$ and $B$, each with $N$ rows (numbered $1$ through $N$) and $N$ columns (numbered $1$ through $N$). Ada may perform the following operation any number of times (including zero): choose an integer $i$ ($1 \le i \le N$), then swap row $i$ and column $i$ in matrix $A$. (Formally, for each valid $j$, she swaps $A_{i, j}$ with $A_{j, i}$.) Is it possible to transform matri