Question

Let $A=\left(\begin{array}{ccc}3& 0& 3\\ 0& 3& 0\\ 3& 0& 3\end{array}\right).$ Then the roots of the equation det (A - $\lambda I_3$) = 0 (where $I_3$ is the identity matrix of order 3) are

• A. 3, 0, 3
• B. 0, 3, 6
• C. 1, 0, -6
• D. 3, 3, 6

Answer 1

Answer: (B)

Let $\left(A-\lambda I_3\right)=0\Rightarrow \left|\begin{array}{ccc}3-\lambda & 0& 3\\ 0& 3-\lambda & 0\\ 3& 0& 3-\lambda \end{array}\right|=0$
$\Rightarrow {\left(3-\lambda \right)}^3-9\left(3-\lambda \right)=0\Rightarrow \left(3-\lambda \right)\left[{\left(3-\lambda \right)}^3-3^z\right]=0$
$\Rightarrow 3-\lambda =0$ or $3-\lambda +3=0\Rightarrow \lambda =0,3$ or $6$