Proof That Similar Matrices Have the Same Characteristic Equation

Theorem
Similar matrices have the same characteristic equation.
Proof
Matrices
$A$
and
$B$
are similar if there is an invertible matrix
$P$
such that
$A=P^{-1}BP$
.
The characteristic equation of the matrix
$A$
is the determinant of the matrix
$A- \lambda I$

\begin{aligned} det(A- \lambda I) &= det(P^{-1}(B- \lambda I)P) \\ &= det(P^{-1}) det (B- \lambda I) det(P) \\ &= \frac{1}{det(P)} det (B- \lambda I) det(P) \\ &= det (B- \lambda I) \end{aligned}

If matrices have the same characteristic solution it does not follow that they are similar. Matrices
$A= \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right) \: B= \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$
both have the characteristic equation
$(1- \lambda )^2$
but these matrices are not similar.