a+babaa+ccbcb+c is equal to

a+babaa+ccbcb+c is equal to

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$|\begin{matrix} a + b & a & b \\ a & a + c & c \\ b & c & b + c \end{matrix}|$ is equal to

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$a^{2} b^{2} c^{2}$

$4 a^{2} b c$

$4 a b c$

$a b c$

answer is A.

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Detailed Solution

$| \begin{matrix} a + b & a & b \\ a & a + c & c \\ b & c & b + c \end{matrix} | R_{1} \to R_{1} - R_{2} - R_{3} | \begin{matrix} 0 & - 2 c & - 2 b \\ a & a + c & c \\ b & c & b + c \end{matrix} | | \begin{matrix} 0 & - 2 c & - 2 b \\ a & a + c & c \\ b & c & b + c \end{matrix} | = 2 c (a b + b^{2} - b c) - 2 b (b c - c^{2} - a c) = 2 a b c + 2 c b^{2} - 2 b c^{2} - 2 b^{2} c + 2 b c^{2} + 2 a b c = 4 a b c$