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