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

The correct option is B $0,3,6$

$det(A-\lambda I_3)=0$
$\Rightarrow |A-\lambda I_3|=0$
$\Rightarrow \left|\begin{array}{ccc}3-\lambda & 0& 3\\ 0& 3-\lambda & 0\\ 3& 0& 3-\lambda \end{array}\right|=0$
$\Rightarrow (3-\lambda )\left[\right(3-\lambda {)}^2-0]+3[0-3(3-\lambda )]=0\Rightarrow (3-\lambda {)}^3-9(3-\lambda )=0$
$\Rightarrow (3-\lambda )\left[\right(3-\lambda {)}^2-3^2]=0\Rightarrow (3-\lambda \left)\right[(6-\lambda )(-\lambda )]=0\Rightarrow \lambda =0,3,6$