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Question: The value of \(\left| \begin{matrix} 1 & 1 & 1 \\ bc & ca & ab \\ b + c & c + a & a + b \end{matrix}...

The value of ∣111bccaabb+cc+aa+b∣\left| \begin{matrix} 1 & 1 & 1 \\ bc & ca & ab \\ b + c & c + a & a + b \end{matrix} \right|is.

A

1

B

0

C

(a−b)(b−c)(c−a)(a - b)(b - c)(c - a)

D

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

Answer

(a−b)(b−c)(c−a)(a - b)(b - c)(c - a)

Explanation

Solution

∣111bccaabb+cc+aa+b∣\left| \begin{matrix} 1 & 1 & 1 \\ bc & ca & ab \\ b + c & c + a & a + b \end{matrix} \right|=∣001c(b−a)a(c−b)abb−ac+aa+b∣\left| \begin{matrix} 0 & 0 & 1 \\ c(b - a) & a(c - b) & ab \\ b - a & c + a & a + b \end{matrix} \right|

{C1→C1−C2,C2→C2−C3}\{ C_{1} \rightarrow C_{1} - C_{2},C_{2} \rightarrow C_{2} - C_{3}\}

= (b−a)(c−b)∣001caab11a+b∣(b - a)(c - b)\left| \begin{matrix} 0 & 0 & 1 \\ c & a & ab \\ 1 & 1 & a + b \end{matrix} \right|= (b−a)(c−a)(c−a)(b - a)(c - a)(c - a)

=(a−b)(b−c)(c−a)= (a - b)(b - c)(c - a).