Q105 / 180 · KCET 2018 · Mathematics
The value of determinant \( \left|\begin{array}{ccc}a-b & b+c & a \\ b-a & c+a & b \\ c-a & a+b & c\end{array}\right| \) is
- A\( a^{3}+b^{3}+c^{3} \)
- B3abc
- C\( a^{3}+b^{3}+c^{3}-3 a b c \)
- Correct answerDNone of the above
Solution
Given that, \( \left|\begin{array}{ccc}a-b & b+c & a \\ b-c & c+a & b \\ c-a & a+b & c\end{array}\right| \)
\( C_{3} \rightarrow C_{2}+C_{3} \)
\( \left|\begin{array}{lll}a-b & b+c & a+b+c \\ b-c & c+a & a+b+c \\ c-a & a+b & a+b+c\end{array}\right| \)
\( =a+b+c)\left|\begin{array}{ccc}a-b & b+c & 1 \\ b-c & c+a & 1 \\ c-a & a+b & 1\end{array}\right| \)
\( =(a+b+c)((c+a)(c-a)-(a+b)(b-c) \)
\( -(b+c)(c-a)-(a+b)(a-b) \)
\( +(b+c)(b-c)-(c+a)(a-b)) \)
\( =(a+b+c)\left(c^{2}-a^{2}-a b+a c-b^{2}+b c\right) \)
\( -b c+a b-c^{2}+a c-a^{2}+b^{2} \)
\( \left.+b^{2}-c^{2}-a c+b c-a^{2}+a b\right) \)
\( =(a+b+c)\left(-c^{2}-3 a^{2}+a b+a c+b^{2}+b c\right) \)
\( =a\left(-c^{2}-3 a^{2}+a b+a c+b^{2}+b c\right) \)
\( +b\left(-c^{2}-3 a^{2}+a b+a c+b^{2}+b c\right) \)
\( +c\left(-c^{2}-3 a^{2}+a b+a c+b^{2}+b c\right) \)
\( =-a c^{2}-3 a^{3}+a^{2} b+a^{2} c+a b^{2}+a b c \)
\( -b c^{2}-3 a^{2} b+a b^{2}+a b c+b^{3}+b^{2} c \)
\( -c^{3}-3 a^{2} c+a b c+a c^{2}+b^{2} c+b c^{2} \)
\( C_{3} \rightarrow C_{2}+C_{3} \)
\( \left|\begin{array}{lll}a-b & b+c & a+b+c \\ b-c & c+a & a+b+c \\ c-a & a+b & a+b+c\end{array}\right| \)
\( =a+b+c)\left|\begin{array}{ccc}a-b & b+c & 1 \\ b-c & c+a & 1 \\ c-a & a+b & 1\end{array}\right| \)
\( =(a+b+c)((c+a)(c-a)-(a+b)(b-c) \)
\( -(b+c)(c-a)-(a+b)(a-b) \)
\( +(b+c)(b-c)-(c+a)(a-b)) \)
\( =(a+b+c)\left(c^{2}-a^{2}-a b+a c-b^{2}+b c\right) \)
\( -b c+a b-c^{2}+a c-a^{2}+b^{2} \)
\( \left.+b^{2}-c^{2}-a c+b c-a^{2}+a b\right) \)
\( =(a+b+c)\left(-c^{2}-3 a^{2}+a b+a c+b^{2}+b c\right) \)
\( =a\left(-c^{2}-3 a^{2}+a b+a c+b^{2}+b c\right) \)
\( +b\left(-c^{2}-3 a^{2}+a b+a c+b^{2}+b c\right) \)
\( +c\left(-c^{2}-3 a^{2}+a b+a c+b^{2}+b c\right) \)
\( =-a c^{2}-3 a^{3}+a^{2} b+a^{2} c+a b^{2}+a b c \)
\( -b c^{2}-3 a^{2} b+a b^{2}+a b c+b^{3}+b^{2} c \)
\( -c^{3}-3 a^{2} c+a b c+a c^{2}+b^{2} c+b c^{2} \)
Hence, the correct answer is option: 'D'.