Question

The vector $\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]$ is an eigen vector of $A=\left[\begin{array}{ccc}-2& 2& -3\\ 2& 1& -6\\ -1& -2& 0\end{array}\right]$. The corresponding eigen value of $A$ is _____.

• A. $1$
• B. $2$
• C. $5$
• D. $-1$

Answer 1

The correct option is C $5$

If ${\lambda }_i$ be the eigen value of matrix $A$ and $X_i$ the corresponding eigen vector then.
$A{X}_i={\lambda }_i{X}_i$
$\Rightarrow L.H.S=\left[\begin{array}{ccc}-2& 2& -3\\ 2& 1& -6\\ -1& -2& 0\end{array}\right]\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]=\left[\begin{array}{c}5\\ 10\\ -5\end{array}\right]=5\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]$
= ${\lambda }_i{X}_i=RHS$
$\Rightarrow {\lambda }_i=5$ is the eigen value of $A$ corresponding to the eigen vector $[12-1{]}^T$