<s>[INST] Confusion on Eigenvalues of Similar Matrices 
Please help me to identify where I went wrong:

The completely reduced normal form of the real matrix $A=
        $\begin{pmatrix}
        1 & 1 & 1 \\
        1 & 1 & 1 \\
        1 & 1 & 1 \\
        \end{pmatrix}
is the following matrix $B=
        $\begin{pmatrix}
        1 & 0 & 0 \\
        0 & 0 & 0 \\
        0 & 0 & 0 \\
        \end{pmatrix}
Clearly the eigenvalues of $B$ are $1,0,0$ so should the eigenvalues of $A$ since similar matrices have the same characteristic polynomial. But when I'm trying to formally evaluate the eigenvalues of $A$ the roots of $\chi_A$ become $0,0,3$.
        $$\begin{vmatrix}
        1-x & 1 & 1 \\
        1 & 1-x & 1 \\
        1 & 1 & 1-x \\
        \end{vmatrix}=0$$
$\implies (1-x)[(1-x)^2-1]-1[(1-x)-1]+1[1-(1-x)]=0$
$\implies (1-x)[x^2-2x]-1(-x)+x=0$
$\implies (1-x)[x^2-2x]+2x=0$
$\implies x^2-2x-x^3+2x^2+2x=0$
$\implies 3x^2-x^3=0$
$\implies x^2(3-x)=0$
So the eigenvalues of $A$ are $0,0,3.$
 [/INST] Your reduced normal form for the matrix A is wrong. 
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