# Show that A and B are not similar matrices A=begin{bmatrix}1 & 0 &1 2 & 0 &2 3&0&3 end{bmatrix} , B=begin{bmatrix}1 & 1 &0 2 & 2 &0 0&1&1 end{bmatrix}

Show that A and B are not similar matrices
$A=\left[\begin{array}{ccc}1& 0& 1\\ 2& 0& 2\\ 3& 0& 3\end{array}\right],B=\left[\begin{array}{ccc}1& 1& 0\\ 2& 2& 0\\ 0& 1& 1\end{array}\right]$
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d2saint0
Step 1
If the matrices A and B are not similar, then their eigenvalues are not the same.
Step 2
Consider the matrix A and find its eigenvalues as follows.
$A=\left[\begin{array}{ccc}1& 0& 1\\ 2& 0& 2\\ 3& 0& 3\end{array}\right]$
$|A-\lambda I|=0$
$|\begin{array}{c}\left[\begin{array}{ccc}1& 0& 1\\ 2& 0& 2\\ 3& 0& 3\end{array}\right]-\lambda \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}|=0$
$|\begin{array}{ccc}1-\lambda & 0& 1\\ 2& -\lambda & 2\\ 3& 0& 3-\lambda \end{array}|=0$
$\left(1-\lambda \right)\left(-\lambda \right)\left(3-\lambda \right)-0+1\left(0+3\lambda \right)=0$
$-{\lambda }^{3}+4{\lambda }^{2}=0$
$\lambda \left(-{\lambda }^{2}+4\lambda \right)=0$
$\lambda \left({\lambda }^{2}-4\lambda \right)=0$
$\lambda \left(\lambda \right)\left(\lambda -4\right)=0$
${\lambda }_{1}=0,{\lambda }_{2}=0,{\lambda }_{3}=4$
Consider the matrix B and find its eigenvalues as follows.
$B=\left[\begin{array}{ccc}1& 1& 0\\ 2& 2& 0\\ 0& 1& 1\end{array}\right]$
$|B-\lambda I|=0$
$|\begin{array}{c}\left[\begin{array}{ccc}1& 1& 0\\ 2& 2& 0\\ 0& 1& 1\end{array}\right]-\lambda \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}|=0$
$|\begin{array}{ccc}1-\lambda & 1& 0\\ 2& 2-\lambda & 0\\ 0& 1& 1-\lambda \end{array}|=0$
$\left(1-\lambda \right)\left(2-\lambda \right)\left(1-\lambda \right)-1\left(2\left(1-\lambda \right)\right)+0=0$
$-{\lambda }^{3}+4{\lambda }^{2}-3\lambda =0$
$\lambda \left(-{\lambda }^{2}+4\lambda -3\right)=0$
$\lambda \left({\lambda }^{2}-4\lambda +3\right)=0$
$\lambda \left(\lambda -1\right)\left(\lambda -3\right)=0$
${\lambda }_{1}=0,{\lambda }_{2}=1,{\lambda }_{3}=3$
Observe that the eigenvalues of the matrices A and B are not the same and thus these matrices are not similar.
Jeffrey Jordon