Suppose $a_{1}, a_{2}, ...........$ real numbers, with $a_{1} \neq 0.$ If $a_{1}, a_{2}, ...........$ are in A.P, then

Easy

A

$A = [\begin{matrix} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}]$ is singular
B

The system of equations $a_{1} x + a_{2} y + a_{3} z = 0, a_{4} x + a_{5} y + a_{6} z = 0, a_{7} x + a_{8} y + a_{8} z = 0$ has infinite number of solutions
C

$B = [\begin{matrix} a_{1} & i a_{2} \\ i a_{2} & a_{1} \end{matrix}]$ is non-singular ; where $i = \sqrt{- 1}$
D

None of these

Solution

We have, $| A | = |\begin{matrix} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}| = |\begin{matrix} a_{1} & a_{2} & a_{3} \\ 3 d & 3 d & 3 d \\ d & d & d \end{matrix}| = 0$
        $[Using R_{3} \to R_{3} - R_{2} a n d R_{2} \to R_{2} - R_{1}]$
$∴$    The given system of homogenous equations has infinite number of solutions.
Also, $| B | = a_{1}^{2} + a_{2}^{2} \neq 0.$
Thus, $B$ is non-singular.

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