Question

Let $A=\left[a_{i j}\right]_{3 \times 3}$ be a scalar matrix whose elements are the roots of the equation $x^{9}-15x^{8}+75x^{7}-125x^{6}=0$ . If $\left|A \cdot a d j A\right|=k$ , then the value of $k$ is equal to

• A. $5^{12}$
• B. $5^{9}$
• C. $3^{12}$
• D. $3^{9}$

Answer 1

Answer: (B)

$x^{9}-15x^{8}+75x^{7}-125x^{6}=0\Rightarrow x^{6}\left(\right. x - 5 \left.\right)^{3}=0$
$\Rightarrow x=0,0,0,0,0,0,5,5,5$
So, $A=\begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$ and $\left|A\right|=5^{3}$
$\left|A \cdot a d j A\right|=\left|\left|A\right| I\right|=\left(\left|A\right|\right)^{3}=\left(5^{3}\right)^{3}=5^{9}$