Question

Let $A = \begin{pmatrix}3&0&3\\ 0&3&0\\ 3&0&3\end{pmatrix}.$ 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 $\left(A - \lambda I_{3}\right) = 0\quad\quad\Rightarrow \begin{vmatrix}3-\lambda&0&3\\ 0&3-\lambda&0\\ 3&0&3-\lambda\end{vmatrix} = 0$
$\Rightarrow \,\left(3-\lambda\right)^{3} - 9\left(3-\lambda\right) = 0\quad\Rightarrow \left(3-\lambda \right)\left[\left(3- \lambda \right)^{3} - 3^{z}\right] = 0$
$\Rightarrow \,3-\lambda = 0$ or $3-\lambda+3 = 0\quad\quad\Rightarrow \lambda = 0, 3$ or $6$