Let $\lambda$ be real. Find all values of $\lambda$ for which the determinant of the matrix $A\lambda I_3$ is zero, where
$$A= \left [ \begin{matrix} 1 & 2 & 1 \\ 0 & 2 & 1 \\ 1 & 0 & 1 \end{matrix} \right ]. $$
I have calculated that $\det(A) = 2$ and $\det(I_3) = 1$ so $\det( 2\lambda 1 ) = 0$.
How to find all values of $\lambda$ ?
As you have already calculated, $\det(A) = 2$. We also know that $\det(\lambda I_3) = \lambda^3$, so by the multiplicativity of the determinant one has
$$\det(A \lambda I_3) = \det(A) \det( \lambda I_3) = 2 \lambda^3.$$
This expression is zero if and only if $\lambda = 0$.