Question

The eigen values of the matrix $\left[\begin{array}{cccc}2& -1& 0& 0\\ 0& 3& 0& 0\\ 0& 0& -2& 0\\ 0& 0& -1& 4\end{array}\right]$ are

• A. $2,-2,1,-1$
• B. $2,3,-2,4$
• C. $2,3,1,4$
• D. None

Answer 1

The correct option is B $2,3,-2,4$

Characteristic equation is
$|A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{cccc}2-\lambda & -1& 0& 0\\ 0& 3-\lambda & 0& 0\\ 0& 0& -2-\lambda & 0\\ 0& 0& -1& 4-\lambda \end{array}\right|=0$
$(2-\lambda )\left|\begin{array}{ccc}3-\lambda & 0& 0\\ 0& -2-\lambda & 0\\ 0& -1& 4-\lambda \end{array}\right|=0$
$(2-\lambda )(3-\lambda )(-2-\lambda )(4-\lambda )=0$
$\lambda =2,3,-2,4$
Hence eigen values are $2,3,-2,4$