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Consider a $3\times 3$ matrix with every element being equal to 1 , its only non-zero eigenvalue is ....?

 \begin{bmatrix} 1 & 1 &1 \\ 1 &1 &1 \\ 1 &1 &1 \end{bmatrix}$

now i solve in simple way ..directly $\left | A-\lambda I \right |$ THEN I got correct ans 0,0,5 but

i have doubt .. if i solve by some elementary operation and reduce to $\begin{bmatrix} 1 & 1 &1 \\ 0& 0 &0 \\ 0 & 0 &0 \end{bmatrix}$

but i got different eigen values  why this happened ... i missed something ??

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