To Find Eigenvalues Find the eigenvalues of the $6\times 6$ matrix
$$\left[\begin{matrix} 
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
\end{matrix}\right]$$
The options  are $1, -1, i, -i$
It is a real symmetric matrix and the eigenvalues of a real symmetric matrix are real. Hence $i$ and $-i$ can't be its  eigenvalues. Then what else we can say?
Is there any easy way to find it?

Note that , $$A=\left[\begin{matrix}O&I\\I&O\end{matrix}\right].$$where,  $I_{3\times 3}$ and $O_{3\times 3}$ are Identity matrix and Zero matrix respectively.
Now, $AA^T=A^2=I_{6\times 6}\implies A$ is orthogonal. So, eigen values are either $1$ or $-1$. Again, $det(A)=-1$.
If all eigen values are $1$ then $det(A)=1$ , contradiction. Again if all eigen values are $-1$ then $det(A)=1$ ,again a contradiction.

Hence , $1$ and $-1$ both are eigen values of $A$.

