1.

If a, b, c are positive and unequal, show that value of the determinant `Delta=|(a,b,c),(b,c,a),(c,a,b)|` is negative

Answer» The given determinant
`=|{:(a,b,c),(b,c,a),(c,a,b):}|`
`=|{:(a+b+c,b,c),(a+b+c,c,a),(a+b+c,a,b):}|" "[C_(1) to (C_(1) + C_(2) + C_(3))]`
`=(a+b+c)*|{:(1,b,c),(1,c,a),(1,a,b):}|`
`=(a+b+c)*|{:(1,b,c),(0,c-b,a-c),(0,a-b,b-c):}| [R_(2) to (R_(2)-R_(1)) " and "R_(3) to (R_(3) -R_(1))]`
`=(a+b+c0 * [(c-b)(b-c)-(a-b)(a-c)] " " ["expanded by"C_(1)]`
`(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)],` which is negative
`[because (a+b+c) gt 0, (a-b)^(2) gt 0, (b-c)^(2) gt 0 " and " (c-a)^(2) gt 0]`


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