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EEJ MAIN Mathematics QUESTION #6016
Question 1

Find all values of $\lambda$ for which the system $2x_1-2x_2+x_3=\lambda x_1$, $\ 2x_1-3x_2+2x_3=\lambda x_2$, $\ {-x_1+2x_2=\lambda x_3}$ has a non-trivial solution.

  • Empty set
  • Singleton set
  • Contains exactly two elements✔️
  • Contains more than two elements
Correct Answer Explanation

Rewrite as $(A-\lambda I)X=0$. For non-trivial solutions, $\det(A-\lambda I)=0$.

$A = \begin{pmatrix}2&-2&1\\2&-3&2\\-1&2&0\end{pmatrix}$

The characteristic equation simplifies to $\lambda^3 + \lambda^2 - 5\lambda + 3 = 0$, which factors as $(\lambda-1)^2(\lambda+3)=0$.

Roots: $\lambda=1$ (repeated) and $\lambda=-3$. So the set contains exactly two elements: $\{1, -3\}$.