The value of the determinant $\mathrm{\Delta} = \left| \begin{matrix} 2a_{1}b_{1} & a_{1}b_{2} + a_{2}b_{1} & a_{1}b_{3} + a_{3}b_{1} \ a_{1}b_{2} + a_{2}b_{1} & 2a_{2}b_{2} & a_{2}b_{3} + a_{3}b_{2} \ a_{1}b_{3} + a_{3}b_{1} & a_{3}b_{2} + a_{2}b_{3} & 2a_{3}b_{3} \end{matrix} ight|$ is

Question

The value of the determinant $\mathrm{\Delta} = \left| \begin{matrix} 2a_{1}b_{1} & a_{1}b_{2} + a_{2}b_{1} & a_{1}b_{3} + a_{3}b_{1} \ a_{1}b_{2} + a_{2}b_{1} & 2a_{2}b_{2} & a_{2}b_{3} + a_{3}b_{2} \ a_{1}b_{3} + a_{3}b_{1} & a_{3}b_{2} + a_{2}b_{3} & 2a_{3}b_{3} \end{matrix} ight|$ is

Dr. Abubacker · Accepted Answer

(c)
We have,
$\mathrm{\Delta} = \left| \begin{matrix}
2a_{1}b_{1} & a_{1}b_{2} + a_{2}b_{1} & a_{1}b_{3} + a_{3}b_{1} \
a_{1}b_{2} + a_{2}b_{1} & 2a_{2}b_{2} & a_{2}b_{3} + a_{3}b_{2} \
a_{1}b_{3} + a_{3}b_{1} & a_{3}b_{2} + a_{2}b_{3} & 2a_{3}b_{3}
\end{matrix} ight|$
This can be written as
$\left| \begin{matrix}
a_{1} & b_{1} & 0 \
a_{2} & b_{2} & 0 \
a_{3} & b_{3} & 0
\end{matrix} ight|\left| \begin{matrix}
b_{1} & a_{1} & 0 \
b_{2} & a_{2} & 0 \
b_{3} & a_{3} & 0
\end{matrix} ight| = 0$

Q. The value of the determinant $\mathrm{\Delta} = \left| \begin{matrix} 2a_{1}b_{1} & a_{1}b_{2} + a_{2}b_{1} & a_{1}b_{3} + a_{3}b_{1} \\ a_{1}b_{2} + a_{2}b_{1} & 2a_{2}b_{2} & a_{2}b_{3} + a_{3}b_{2} \\ a_{1}b_{3} + a_{3}b_{1} & a_{3}b_{2} + a_{2}b_{3} & 2a_{3}b_{3} \end{matrix} \right|$ is

Explanation

Reviewed by Dr. Abubacker

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