Question

The eigen values of the matrix given below are: $\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& -3& -4\end{array}\right]$

• A. (0, -1, -3)
• B. (0, -2, -3)
• C. (0, 2, 3)
• D. (0, 1, 3)

Answer 1

The correct option is A (0, -1, -3)

A = $\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& -3& -4\end{array}\right]$

Characteristic equation is $|A-\lambda I|$ = 0

$\left|\begin{array}{ccc}-\lambda & 1& 0\\ 0& -\lambda & 1\\ 0& -3& (-4-\lambda )\end{array}\right|$ = 0

$\Rightarrow -\lambda \left[\right(-\lambda \left)\right(-4-\lambda )-(-3\left)\right]-1\left[0\right]+0=0$

$\Rightarrow$ ${\lambda }^2(4+\lambda )+3\lambda$ = 0

$\Rightarrow$ ${\lambda }^3+4{\lambda }^2+3\lambda$ = 0

$\Rightarrow$ $\lambda ({\lambda }^2+4\lambda +3)$ = 0

$\lambda$ = (0, -1, -3)
So eigen values are $0,-1,-3$