Question

The determinant $\left| \begin{matrix} 4 + x^{2} & - 6 & - 2 \

• 6 & 9 + x^{2} & 3 \
• 2 & 3 & 1 + x^{2} \end{matrix} \right|$ is not divisible by.

• A. x
• B. $x^{3}$
• C. $14 + x^{2}$
• D. $x^{5}$

Answer 1

Answer: (D)

$x^{5}$

Answer 2

$\left| \begin{matrix} 4 + x^{2} & - 6 & - 2 \

• 6 & 9 + x^{2} & 3 \
• 2 & 3 & 1 + x^{2} \end{matrix} \right| = x^{4}(14 + x^{2}) = x.x^{3}(14 + x^{2})$

Hence, the determinant is divisible by x,$x^{3}$ and $(14 + x^{2})$, but not divisible by $x^{5}$.