Question

# The value of the determinant $$\displaystyle \left | \begin{matrix}b+c &a-b &a \\ c+a& b-c &b \\ a+b& c-a & c\end{matrix} \right |$$ , is

A
a3+b3+c33abc
B
3abca3b3c3
C
3abc+a3+b3+c3
D
none of these

Solution

## The correct option is B $$\displaystyle 3abc-a^{3}-b^{3}-c^{3}$$$$\left| \begin{matrix} b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c \end{matrix} \right|$$$${ C }_{ 1 }\rightarrow { C }_{ 1 }+{ C }_{ 2 }+{ C }_{ 3 }$$$$\left| \begin{matrix} 2(a+b+c) & 0 & a+b+c \\ c+a & b-c & b \\ a+b & c-a & c \end{matrix} \right|$$$$=(a+b+c)\left| \begin{matrix} 2 & 0 & 1 \\ c+a & b-c & b \\ a+b & c-a & c \end{matrix} \right|$$$${ C }_{ 1 }\rightarrow { C }_{ 1 }-{ 2C }_{ 3 }$$$$=(a+b+c)\left| \begin{matrix} 0 & 0 & 1 \\ c+a-2b & b-c & b \\ a+b-2c & c-a & c \end{matrix} \right|$$$$=(a+b+c)[(c-a)(c+a-2b)-(b-c)(a+b-2c)]=(a+b+c)[{ -a }^{ 2 }-{ b }^{ 2 }{ -c }^{ 2 }+ab+bc+ac]$$$$=3abc-{ a }^{ 3 }{ -b }^{ 3 }{ -c }^{ 3 }$$Mathematics

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