Let a , b and c be positive and not all equal. Show that the value of the determinant `|{:(a,,b,,c),(b ,,c,,a),(c,,a,,b):}|` is negative .

Options

A. `+ ive`
B. `-ive`
C. zero
D. none of these

Answer

B

Explanation

Let `Delta = `|(a,b,c),(b,c,a),(c,a,b)|`. Then,
`Delta = `|(a + b + c,b,c),(b + c + a,c,a),(c + a + b,a,b)|" " ["Applying " C_(1) rarr C_(1) + C_(2) + C_(3)]`
`rArr Delta = (a + b + c) |(1,b,c),(1,c,a),(1,a,b)| " " ["Taking " (a +b +c) " common " C_(1)]`
`rArr Delta = (a +b +c) |(1,b,c),(0,c -a,a -c),(0,a -b,b -c)| [("Applying " R_(2) rarr R_(2) - R_(1)),(" " R_(3) rarr R_(3) - R_(1))]`
`rArr Delta = (a +b +c) 1 |(c -b,a -c),(a-b,b -c)| " " ["Expanding along " C_(1)]`
`rArr Delta = (a + b +c) (-(b -c)^(2) -(a -c) (a -b))`
`rArr Delta = - (a +b + c) [(b -c)^(2) + (a -c) (a -b)]`
`rArr Delta = -(1)/(2) ( a +b +c) {(a -b)^(2) + (b -c)^(2) + (c -a)^(2)} lt 0 " " [ :' a != b != and a, b, c gt 0]`