Question

For a given matrix $A=\left[\begin{array}{ccc}2& -2& 3\\ -2& -1& 6\\ 1& 2& 0\end{array}\right]$, one of the eigen values is 3. The other two eigen values are.

• A. $2,-5$
• B. $3,-5$
• C. $2,5$
• D. $3,5$

Answer 1

The correct option is B $3,-5$

$A=\left[\begin{array}{ccc}2& -2& 3\\ -2& -1& 6\\ 1& 2& 0\end{array}\right]$,
Trick: Let other eigen values are ${\lambda }_1,{\lambda }_2$,
Now,
$3+{\lambda }_1+{\lambda }_2=2+(-1)+0=1$
${\lambda }_1+{\lambda }_2=-2$ .... (1)
Check options, only (b) satisfies (1)
Hence eigen values are $3,-5$