If $\left| \begin{matrix} a_{1} & b_{1} & c_{1} \ a_{2} & b_{2} & c_{2} \ a_{3} & b_{3} & c_{3} \end{matrix} ight| = 5,$ then the value of $\left| \begin{matrix} b_{2}c_{3} - b_{3}c_{2} & c_{2}a_{3} - c_{3}a_{2} & a_{2}b_{3} - c_{3}b_{2} \ b_{3}c_{1} - b_{1}c_{3} & c_{3}a_{1} - c_{1}a_{3} & a_{3}b_{1} - a_{1}b_{3} \ b_{1}c_{2} - b_{2}c_{1} & c_{1}a_{2} - c_{2}a_{1} & a_{1}b_{2} - a_{2}b_{1} \end{matrix} ight|$ is

Question

If $\left| \begin{matrix} a_{1} & b_{1} & c_{1} \ a_{2} & b_{2} & c_{2} \ a_{3} & b_{3} & c_{3} \end{matrix} ight| = 5,$ then the value of $\left| \begin{matrix} b_{2}c_{3} - b_{3}c_{2} & c_{2}a_{3} - c_{3}a_{2} & a_{2}b_{3} - c_{3}b_{2} \ b_{3}c_{1} - b_{1}c_{3} & c_{3}a_{1} - c_{1}a_{3} & a_{3}b_{1} - a_{1}b_{3} \ b_{1}c_{2} - b_{2}c_{1} & c_{1}a_{2} - c_{2}a_{1} & a_{1}b_{2} - a_{2}b_{1} \end{matrix} ight|$ is

Dr. Abubacker · Accepted Answer

(b)
Let $A \equiv \left| \begin{matrix}
a_{1} & b_{1} & c_{1} \
a_{2} & b_{2} & c_{2} \
a_{3} & b_{3} & c_{3}
\end{matrix} ight| = 5$ ...(i)
$	herefore\ \left| \begin{matrix}
b_{2}c_{3} - b_{3}c_{2} & c_{2}a_{3} - c_{3}a_{2} & c_{2}b_{3} - c_{3}b_{2} \
b_{3}c_{1} - b_{1}c_{3} & c_{3}a_{1} - c_{1}a_{3} & a_{3}b_{1} - a_{1}b_{3} \
b_{1}c_{2} - b_{2}c_{1} & c_{1}a_{2} - c_{2}a_{1} & a_{1}b_{2} - a_{2}b_{1}
\end{matrix} ight|$
$|adj\ A| = (5)^{3 - 1}$ [from Eq. (i)]
$= 5^{2} = 25\ \ \ (\because|adj\ A| = |A|^{n - 1})$

Explanation

Reviewed by Dr. Abubacker

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