<s>[INST] Show that a given Eigenvalue belongs to Matrix without characteristic equation How do I show that the given a eigenvalue $λ=1$ is an eigenvalue to matrix
$$ T=\begin{bmatrix}-11 &9& 6\\ -8& 6& 2\\ -6& 6& 7\end{bmatrix}$$
WITHOUT using the characteristic equation... I have NO clue, please help with this :)
 [/INST] Consider the matrix 
$$ T - \lambda = T- 1 = \begin{pmatrix} -12 & 9 & 6 \\ -8 & 5 & 2 \\ -6 & 6 & 6 \end{pmatrix} $$
Now transform it to row echelon form to get 
$$ \begin{pmatrix} -12 & 9 & 6 \\ 0 & -1 & -2 \\ 0 & \frac 32 & 3 \end{pmatrix} \leadsto \begin{pmatrix} -12 & 9 & 6 \\ 0 & -1 & -2 \\ 0 & 0 & 0 \end{pmatrix}$$
As $T-1$ has rank $2$, $\dim\ker (T- 1) = 1$ and $\lambda = 1$ is an eigenvalue.
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