Question

The characteristic roots of the matrix $\left[\begin{array}{ccc}1& 0& 0\\ 2& 3& 0\\ 4& 5& 6\end{array}\right]$ are.

• A. $1,3,6$
• B. $1,2,4$
• C. $4,5,6$
• D. $2,4,6$

Answer 1

Answer: (A)

$1,3,6$

Characteristic roots of a matrix $A$ is $\lambda$ where $\lambda$ is $|A-I\lambda |=0$ and I is identity matrice.

$|A-I\lambda |$

$\Rightarrow |\left(\begin{array}{ccc}1& 0& 0\\ 2& 3& 0\\ 4& 5& 6\end{array}\right)-(\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\times \lambda )|=0\Rightarrow \left|\left(\begin{array}{ccc}1-\lambda & 0& 0\\ 2& 3-\lambda & 0\\ 4& 5& 6-\lambda \end{array}\right)\right|=0\Rightarrow 1-\lambda \left(\right(3-\lambda )\times (6-\lambda )-5\times 0)-0+0=0\Rightarrow (1-\lambda )(3-\lambda )(6-\lambda )=0\Rightarrow \lambda =1,3,6$