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Let a , b and c be positive and not ...

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 .

Text Solution

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`Delta=|{:(a,b,c),(v,c,a),(c,a,b):}|C_1toC_1+C_2+C_3`
`Delta=|{:(a+b+c,b,c),(B+c+a,c,a),(c+a+b,a,b):}|=(a+b+c)|{:(1,b,c),(1,c,a),(1,a,c):}|R_2toR_2-R_1,R_3toR_3-R_1`
`=(a+b+c)|{:(1,b,c),(0,c-b,a-c),(0,a-b,b-c):}|`
Expanding along `C_1`
`=(a+b+c){(c-a)(b-c)-(a-b)(a-c)}`
`=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)`
`=-(a+b+c)1/2{(a-b)^2+(b-c)^2+(c-a)^2}`
As a, b, c is positive hence `Delta` is negative .
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