Working out eigenvalues of this matrix without any calculations

Question

$$
\begin{pmatrix}
110 & 55 & -164 \\
42 & 21 & -62 \\
88 & 44 & -131 \\
\end{pmatrix}
$$

This is the matrix. I have worked out so far that because the columns are linearly dependent, then one of the eigenvalues is $0$. The sum of the diagonal values is $110+21+(-131)=0$ and this is the sum of the eigenvalues. So I know the other two eigenvalues are $a$ and $-a$. How would I work out $a$ without writing down any calculations? Any hints/tips are greatly appreciated!

I also notice that the 3rd column is very close to being the negative sum of the first two columns.

Thanks for your time 🙂

Answer 1

What do you count as "calculation"? Note that the row sums are all equal to $1$. Hence, $(1,1,1)^T$ is an eigenvector with eigenvalue $1$. Since the trace is $0$ the remaining eigenvalue must be $-1$.