Let a, b, 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

Explanation

` Delta =|{:(a,,b,,c),(b ,,c,,a),(c,,a,,b):}|`
Expanding using Sarrus rule we get
`Delta =abc + abc +abc -a^(3) -b^(3)-c^(3)`
`=- (a^(3) =b^(3) +c^(3) - 3abc)`
`=-(a+b+c ) [a^(2) +b^(2) +c^(2) -ab -bc -ca]`
` =-(1)/(2) (a+b+c) [2a^(2) + 2b^(2) + 2c^(2) - 2ab- 2bc - 2ca]`
`=-(1)/(2) (a + b+ c)[(a-b)^(2) + (b -c)^(2) + (c-a)^(2)]`
Since a,b,c ` gt 0 , a + b+ c gt0`
Also ,` a ne b ne c`
`:. Delta lt 0`