Answer
$-8-2\lambda+5\lambda^2-\lambda^3$
Work Step by Step
We need to find the determinant of $\begin{bmatrix}
-1-\lambda&0&1\\
-3&4-\lambda&1\\
0&0&2-\lambda
\end{bmatrix}$
$(-1)^{3+3}(2-\lambda)\left|
\begin{matrix}
-1-\lambda&0\\
-3&4-\lambda
\end{matrix}
\right|=(2-\lambda)\big((-1-\lambda)(4-\lambda)\big)=(2-\lambda)(-4-3\lambda+\lambda^2)=-8-2\lambda+5\lambda^2-\lambda^3$
