Question

The three characteristic roots of the following matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 2& 3\\ 0& 0& 2\end{array}\right]$ are

• A. $1,2,3$
• B. $1,2,2$
• C. $1,0,0$
• D. $0,2,3$

Answer 1

The correct option is B $1,2,2$

Characteristic equation of A is
$|A-\lambda I|=0$
$\left[\begin{array}{ccc}1-\lambda & 2& 3\\ 0& 2-\lambda & 3\\ 0& 0& 2-\lambda \end{array}\right]=0$
$(1-\lambda )(2-\lambda {)}^2=0$
$\lambda =1,2,2$
Alternate:
Given matrix is upper triangular matrix.
Eigen values of a upper triangular matrix are it's diagonal elements. So eigen values are 1, 2, 2