If a,b,c are comples number and z=`|{:(,0,-b,-c),(,bar(b),0,-a),(,bar(c),bar(a),0):}|` then show tha z is purely imaginary

Options

A. is a non-zero real number
B. purely imaginary
C. 0
D. none of these

Answer

B

Explanation

We observe that
`bar(Delta) = |(0,-bar(b),- bar(c)),(b,0,-bar(a)),(c,a,0)|`
`rArr bar(Delta) = -|(0,bar(b),bar(c)),(-b,0,bar(a)),(-c,-a,0)|` [Taking `(-1)` common from each row]
`rArr bar(Delta) = - |(0,-b,-c),(bar(b),0,-a),(bar(c),bar(a),0)|`[Interchanging rows and columns]
`rArr bar(Delta) = - Delta rArr Delta + bar(Delta) = 0 rArr Delta` is purely imaginary