$if\left| \begin{matrix} 1 + ax & 1 + bx & 1 + cx \ 1 + a_{1}x & 1 + b_{1}x & 1 + c_{1}x \ 1 + a_{2}x & 1 + b_{2}x & 1 + c_{2}x \end{matrix} ight| = A_{0} + A_{1}x + A_{2}x^{2} + A_{3}x^{3},\ then\ A_{0}$ is equal to

Question

$if\left| \begin{matrix} 1 + ax & 1 + bx & 1 + cx \ 1 + a_{1}x & 1 + b_{1}x & 1 + c_{1}x \ 1 + a_{2}x & 1 + b_{2}x & 1 + c_{2}x \end{matrix} ight| = A_{0} + A_{1}x + A_{2}x^{2} + A_{3}x^{3},\ then\ A_{0}$ is equal to

Dr. Abubacker · Accepted Answer

(b)
Given that, $\left| \begin{matrix}
1 + ax & 1 + bx & 1 + cx \
1 + a_{1}x & 1 + b_{1}x & 1 + c_{1}x \
1 + a_{2}x & 1 + b_{2}x & 1 + c_{2}x
\end{matrix} ight|$
$= A_{0} + A_{1}x + A_{2}x^{2} + A_{3}x^{3}$
On putting $x = 0$ on both sides, we get
$\left| \begin{matrix}
1 & 1 & 1 \
1 & 1 & 1 \
1 & 1 & 1
\end{matrix} ight| = A_{0}$
$\Rightarrow A_{0} = 0$

Q. $if\left| \begin{matrix} 1 + ax & 1 + bx & 1 + cx \\ 1 + a_{1}x & 1 + b_{1}x & 1 + c_{1}x \\ 1 + a_{2}x & 1 + b_{2}x & 1 + c_{2}x \end{matrix} \right| = A_{0} + A_{1}x + A_{2}x^{2} + A_{3}x^{3},\ then\ A_{0}$ is equal to

Explanation

Reviewed by Dr. Abubacker

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