Question

If $a_1a_2a_3......a_9$ are in A.P. then the value of
$\begin{vmatrix}a_{1}&a_{2}&a_{3}\\ a_{4}&a_{5}&a_{6}\\ a_{7}&a_{8}&a_{9}\end{vmatrix}$ is

• A. $\frac{9}{2}(a_1+a_9)$
• B. $(a_1+a_9)$
• C. $log_e(log_e e)$
• D. 1

Answer 1

Answer: (C)

Let $d$ be the common difference of A.P.
Then, $ \Delta=\begin{vmatrix} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{vmatrix}=\begin{vmatrix} a_{1} & a_{1}+d & a_{1}+2 d \\ a_{1}+3 d & a_{1}+4 d & a_{1}+5 d \\ a_{1}+6 d & a_{1}+7 d & a_{1}+8 d \end{vmatrix}$ Applying $C_{2} \rightarrow C_{2}-C_{1}$ and $C_{3} \rightarrow C_{3}-C_{2}$
$\Delta =\begin{vmatrix} a_{1} & d & d \\ a_{1}+3 d & d & d \\ a_{1}+6 d & d & d \end{vmatrix} $
$=0$
$\left(\because C_{2}\right.$ and $C_{3}$ are identical)
$=\log _{ e }\left(\log _{e} e\right) \,\,\,\left(\because \log _{e} 1=0\right)$