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Using the factor theorem it is found tha...

Using the factor theorem it is found that a+b , b+c and c+a are three factors of the determinant `|[-2a ,a+b, a+c],[ b+a,-2b,b+c],[c+a ,c+b,-2c]|`. The other factor in the value of the determinant is

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`Delta=k(a+b)(b+c)(c+a)`

`=>|[-2a ,a+b, a+c],[ b+a,-2b,b+c],[c+a ,c+b,-2c]|=k(a+b)(b+c)(c+a)`

`a=0,b=1,c=2`

`=>|[0 ,1, 2],[ 1,-2,3],[2,3,-4]|=k(0+1)(1+2)(2+0)`

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