The-equation-determinant-x-a-x-b-x-c-x-b-x-c-x-a-x-c-x-a-x-b-0-where-a-b-c-are-different-is-satisfied-by-

Question Number 15194 by arnabpapu550@gmail.com last updated on 08/Jun/17

The equation | \begin{matrix} x - a & x - b & x - c \\ x - b & x - c & x - a \\ x - c & x - a & x - b \end{matrix} | = 0,

where a, b, c are different, is satisfied by

Commented by prakash jain last updated on 08/Jun/17

| \begin{matrix} x - a & x - b & x - c \\ x - b & x - c & x - a \\ x - c & x - a & x - b \end{matrix} | = 0,

R 1 = R 1 + R 2 + R 3

| \begin{matrix} 3 x - a - b - c & 3 x - a - b - c & 3 x - c - b - a \\ x - b & x - c & x - a \\ x - c & x - a & x - b \end{matrix} | = 0,

(3 x - a - b - c) | \begin{matrix} 1 & 1 & 1 \\ x - b & x - c & x - a \\ x - c & x - a & x - b \end{matrix} | = 0,

C 1 = C 1 - C 2, C 2 = C 2 - C 3

(3 x - a - b - c) | \begin{matrix} 0 & 0 & 1 \\ c - b & a - c & x - a \\ a - c & b - a & x - b \end{matrix} | = 0,

3 x - a - b - c = 0 \Rightarrow x = \frac{a + b + c}{3}

Commented by arnabpapu550@gmail.com last updated on 08/Jun/17

excellent!