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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  
\[A- \lambda I\]

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

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.

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