If two matrices have the same determinant, are they similar? I am wondering if we have two square matrices $A$ and $B$ and if $\det A = \det B$, then does an invertible matrix $P$ exist with 
$$A = P^{-1} B P$$
 A: No. Take $A=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ and $B=\begin{pmatrix}0&0\\0&0\end{pmatrix}$.
A: Just as two rectangles with the same area are not necessarily similar (and, in fact, won't be unless they are the same or flipped about the 45-degree axis), two matrices with equal determinants will not necessarily be similar. This is not just a coincidence; it actually gets to the concept of the determinant, which you might be interested to explore further.
A: No. Take
$$A=\begin{pmatrix}1&1\\0&1\end{pmatrix}\:\:{\rm and}\:\:B=\begin{pmatrix}1&0\\0&1\end{pmatrix}$$
Can you argue why $A\nsim B$?
A: A good counterexample is:
$$A=\begin{pmatrix}1&0\\0&6\end{pmatrix}\:\:{\rm and}\:\:B=\begin{pmatrix}2&0\\0&3\end{pmatrix},$$
both of which have determinant $6$. Can you see why these matrices are not similar?
A: No, they are not even guaranteed to be the same dimension.  Try 2 identity matrices of different dimensions.  But even if they are the same dimension, try 2 different triangular matrices with different off-diagonal members.
