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Question: \[\left| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} \right| =\]...

1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} \right| =$$
A

a2+b2+c2a^{2} + b^{2} + c^{2}`

B

(a+b)(b+c)(c+a)(a + b)(b + c)(c + a)

C

(a−b)(b−c)(c−a)(a - b)(b - c)(c - a)

D

None of these

Answer

(a−b)(b−c)(c−a)(a - b)(b - c)(c - a)

Explanation

Solution

1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} \right|$$ Applying $R_{1} \rightarrow R_{1} - R_{2}\& R_{2} \rightarrow R_{2} - R_{3}$ $= \left| \begin{matrix} 0 & a - b & a^{2} - b^{2} \\ 0 & b - c & b^{2} - c^{2} \\ 1 & c & c^{2} \end{matrix} \right| = (a - b)(b - c)\left| \begin{matrix} 0 & 1 & a + b \\ 0 & 1 & b + c \\ 1 & c & c^{2} \end{matrix} \right|$; Applying $R_{1} \rightarrow R_{1} - R_{2}$ $$= (a - b)(b - c)\left| \begin{matrix} 0 & 0 & (a - c) \\ 0 & 1 & b + c \\ 1 & c & c^{2} \end{matrix} \right|$$ $$= (a - b)(b - c)(a - c)\left| \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & b + c \\ 1 & c & c^{2} \end{matrix} \right| = (a - b)(b - c)(a - c)( - 1) = (a - b)(b - c)(c - a)$$