Question

The value of the determinant `|(b +c,a -b,a),(c +a,b -c,b),(a +b,c -a,c)|`, is
A. `a^(3) + b^(3) + c^(3) - 3abc`
B. `3abc - a^(3) -b^(3) - c^(3)`
C. `3 abc + a^(3) + b^(3) + c^(3)`
D. none of these

Answer 1

Correct Answer - B
We have,
`|(b +c,a -b,a),(c +a,b -c,b),(a +b,c -a,c)|`
`= |(a +b +c,-b,a),(b +c +a,-c,b),(c +a +b,-a,c)| " " ["Applying " C_(1) rarr C_(1) + C_(3), C_(2) rarr -C_(2) - C_(3)]`
`= - (a +b +c) |(1,b,a),(1,c,a),(1,a,c)|`
`= -(a +b +c) |(1,b,a),(0,c -b,b -a),(0,a -b,c -a)| [(" Applying"),(R_(2) rarr R_(2) - R_(1) ", "R_(3) rarr R_(3) - R_(1))]`
`= - (a + b +c) (a^(2) + b^(2) + c^(2)- ab - bc -ca)`
`= - (a^(3) + b^(3) + c^(3) - 3abc)`