<s>[INST] Eigenvalue of $A-aI_3$ Question: Let $A=\begin{pmatrix}  a+1 & 1 & 1 \\ 1 & a+1 & 1 \\  1 & 1 & a+1\end{pmatrix}$. Show that $A-aI_3$ has eigenvalue of 3. Also find eigenvector.
My thinking: 
I know that we have to apply characteristic polynomial $|A- \lambda I|$ to find the eigenvalue. I don't understand the part $A-aI_3$. What should I suppose to do here with this $A-aI_3$ ? I am kinda confused. Your kind suggestion will be appreciated. 
 [/INST] Alternative answer:  $A-aI_3=\begin{pmatrix}  1 & 1 & 1 \\ 1 & 1 & 1 \\  1 & 1 & 1\end{pmatrix}$ clearly has eigenvalue $0$ 
with independent eigenvectors $\begin{pmatrix}  1  \\ -1  \\  0 \end{pmatrix}$ and $\begin{pmatrix}  1  \\ 0  \\  -1 \end{pmatrix}.$
The sum of the eigenvalues is the trace of the matrix, which is $3$, so the other eigenvalue is $3$.  
An eigenvector corresponding to the eigenvalue $3$ is in the kernel of  $\begin{pmatrix}  -2 & 1 & 1 \\ 1 & -2 & 1 \\  1 & 1 & -2\end{pmatrix}$ , i.e. $(-2,1,1)\times(1,-2,1)=(-3,-3,-3)$.
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