Question

The set of all values of $\lambda$ for which the system of linear equations
$2x_1-2x_2+x_3=\lambda x_1$
$2x_1-3x_2+2x_3=\lambda x_2$
$-x_1+2x_2=\lambda x_3$ has a non-trivial solution

• A. is an empty set
• B. is a singleton
• C. contains two elements
• D. contains more than two elements

Answer 1

Answer: (D)

contains more than two elements

Answer 2

The correct answer is D: contains more than 2 elements
$\Delta=(2-\lambda)\left(\lambda^{2}+3 \lambda-4\right)+2(-2 \lambda+2)+1$
$(4-3-\lambda)=0$
$-\lambda^{3}-\lambda^{2}+6 \lambda+8-3-\lambda-8=0$
$-\lambda^{3}-\lambda^{2}+5 \lambda-3=0$
$\lambda^{3}+\lambda^{2}-5 \lambda+3=0$
$(\lambda-1)\left(\lambda^{2}+2 \lambda-3\right)=0$
$(\lambda-1)(\lambda+3)(\lambda-1)=0$
$\lambda=1,1,-3$