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Question: \[\left| \begin{matrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{matri...

a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{matrix} \right| =$$
A

(a+b+c)2(a + b + c)^{2}

B

(a+b+c)3(a + b + c)^{3}

C

(a+b+c)(ab+bc+ca)(a + b + c)(ab + bc + ca)

D

None of these

Answer

(a+b+c)3(a + b + c)^{3}

Explanation

Solution

∣a−b−c2a2a2bb−c−a2b2c2cc−a−b∣\left| \begin{matrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{matrix} \right|

= $\left| \begin{matrix}

  • \Sigma a & 0 & 2a \ \Sigma a & - \Sigma a & 2b \ 0 & \Sigma a & c - a - b \end{matrix} \right|,,\left( \begin{aligned} & C_{1} \rightarrow C_{1} - C_{2} \ & C_{2} \rightarrow C_{2} - C_{3} \end{aligned} \right)$

= $(\Sigma a)^{2}\left| \begin{matrix}

  • 1 & 0 & 2a \ 1 & - 1 & 2b \ 1 & 1 & c - a - b \end{matrix} \right| = (\Sigma a)^{3}$, (on expansion)

= (a+b+c)3(a + b + c)^{3}.