Q.

Suppose a1,a2,...........  real numbers, with a1≠0.  If a1,a2,...........  are in A.P, then

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a

A=a1a2a3a4a5a6a7a8a9  is singular

b

The system of equations a1x+a2y+a3z=0,  a4x+a5y+a6z=0,  a7x+a8y+a8z=0  has infinite number of solutions

c

B=a1ia2ia2a1  is non-singular ; where i=−1

d

None of these

answer is A.

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Detailed Solution

We have, |A|=a1a2a3a4a5a6a7a8a9=a1a2a33d3d3dddd=0          [Using  R3→R3−R2  and   R2→R2−R1]  ∴     The given system of homogenous equations has infinite number of solutions.Also, |B|=a12+a22≠0.  Thus,B is non-singular.
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Suppose a1,a2,...........  real numbers, with a1≠0.  If a1,a2,...........  are in A.P, then