Let $A,B$ be a $n \times n$ matrices such that $\det(B) = 1$. Will the spectrum
(set of eigenvalues) $AB$ be same as that of $A$. Or, at least is $\mbox{Trace}(A) = \mbox{Trace}(AB)$ ? If not, what can we say about the change in spectrum and trace in $A$ and $AB$.
[Math] Trace and eigenvalues under multiplication by a matrix with determinant $1$
matrices
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Best Answer
If two matrices $A,B$ have the same nonzero determinant, then $\det(A^{-1}B)=1$ so $A$ and $B$ are in the same class under multiplication by matrices of determinant $1$. Hence among invertible matrices the only invariant quantity under such multiplication is the determinant itself (of course any function of the determinant is also invariant; this is not very interesting). So there is no chance that the characteristic polynomial, spectrum, trace or whatever is invariant.