Suppose `a_(1), a_(2)`, .... Are real numbers, with `a_(1) ne 0`. If `a_(1), a_(2), a_(3)`, ... Are in A.P., then

Options

A. `A=[(a_(1),a_(2),a_(3)),(a_(4),a_(5),a_(6)),(a_(5),a_(6),a_(7))]` is singular (where `i=sqrt(-1)`)
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_(9)z=0` has infinite number of solutions
C. `B[(a_(1),ia_(2)),(ia_(2),a_(1))]` is nonsingular
D. none of these

Answer

A, B, C

Explanation

Applying `R_(3) rarr R_(3)-R_(2), R_(2) rarr R_(2)-R_(1)`, we get
`|A|=3|(a_(1),a_(2),a_(3)),(d,d,d),(d,d,d)|=0`
where d is the common difference of the A.P.
Therefore, the given system of equations has infinite number of solutions. Also,
`|B|=a_(1)^(2)+a_(2)^(2) ne 0`