Are the matrices $A = \begin{bmatrix}1&2&-3\\1&-1&2\\0&3&5\end{bmatrix}$ and $B = \begin{bmatrix}1&-1&4\\1&1&5\\1&1&6\end{bmatrix}$ similar?
I know that similar matrices have the same determinant, the same rank, and the same characteristic polynomial (and therefore, the same eigenvalues).
What I have tried:
$det(A) = -30$
$det(B) = 2$
$rank(A) = 3$
$rank(B) = 3$
How do I show the characteristic polynomial for $A$ and $B$?
Also since $det(A) \ne det(B)$, we already know that matrices $A$ and $B$ are not similar so we don't need to prove that using $B = P^{-1}AP$, right?
Best Answer
Easiest property is that similar matrices have the same trace, and not in your case.