The determinant | , * 20 c 4 + x^2 & - 6 & - 2 - 6 & 9 + x^2 &3 -… - Vidyadip

MCQ

The determinant $\left| {\,\begin{array}{*{20}{c}}{4 + {x^2}}&{ - 6}&{ - 2}\\{ - 6}&{9 + {x^2}}&3\\{ - 2}&3&{1 + {x^2}}\end{array}\,} \right|$ is not divisible by

A
$x$
B
${x^3}$
C
$14 + {x^2}$
✓
${x^5}$

✓

Answer

Correct option: D.

${x^5}$

d
(d) $\left| {\,\begin{array}{*{20}{c}}{4 + {x^2}}&{ - 6}&{ - 2}\\{ - 6}&{9 + {x^2}}&3\\{ - 2}&3&{1 + {x^2}}\end{array}} \right| = {x^4}(14 + {x^2})$ $ = x.{x^3}(14 + {x^2})$

Hence, the determinant is divisible by $x$,${x^3}$ and $(14 + {x^2})$,

but not divisible by ${x^5}$.

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