Answer to Question #236806 in Linear Algebra for moe

Question

Find all real values of λ such that 

$\left| \begin{array}{cccc}1 & \lambda & -1\\ 1 & -1 & -\lambda \\ \lambda & -1 & 1\end{array} \right |=0.$

1. λ=0
2. $\lambda=\sqrt{3}$
3. $\lambda=-\sqrt{3}$
4. There is no such λ
5. λ=1

Answer 1

$$\begin{vmatrix} 1 & \lambda & -1 \\ 1 & -1 & -\lambda \\ \lambda & -1 & 1 \\ \end{vmatrix}=1\begin{vmatrix} -1 & -\lambda \\ -1 & 1 \end{vmatrix}-\lambda\begin{vmatrix} 1 & -\lambda \\ \lambda & 1 \end{vmatrix}-1\begin{vmatrix} 1 & -1 \\ \lambda & -1 \end{vmatrix}$$

$$=-1-\lambda-\lambda-\lambda^2+1-\lambda=-\lambda^2-3\lambda=0$$

$$-\lambda(\lambda+3)=0$$

$$\lambda_1=-3, \lambda_2=0$$

1. $\lambda=0$