Question

(xii) \( \left|\begin{array}{ccc}a & b & c \\ a^{2} & b^{2} & c^{2} \\ b+c & c+a & a+b\end{array}\right|=(b-c)(c-a)(a-b)(a+b+c) \)

Answer 1

\(\begin{vmatrix}a&b&c\\a^2&b^2&c^2\\b+&c+a&a+b\end{vmatrix}\)

Applying C₂→C₂→C₁, C₃→C₃ - C₁  

\(=\begin{vmatrix}a&b-a&c-a\\a^2&(b-a)(b+a)&(c-a)(c+a)\\b+c&-(b-a)&-(c-a)\end{vmatrix}\)

Applying C₂→C₂/b-a, C₃→C₃/c-a

 \(=(b-a)(c-a)\begin{vmatrix}a&1&1\\a^2&(b+a)&(c+a)\\b+c&-1&-1\end{vmatrix}\)

Applying C₃→ \(\frac{C_3-C_2}{c-b}\)

\(=(b-a)(c-a)(c-b)\begin{vmatrix}a&1&0\\a^2&(b+a)&1\\b+c&-1&0\end{vmatrix}\)

on expanding determinant along column C₃

= (a - b) (b -c) (c - a) (-1)\(\begin{vmatrix}a& 1\\b+c&-1\end{vmatrix}\)

= (a - b) (b - c) (c - a) (-1) (-a  - b - c)

 = (a - b) (b - c) (c - a) (a + b +c)