The value of the determinant $\begin{vmatrix} a-b & b+c & a\\ b-c & c+a & b \\ c-a & a+ b & c\end {vmatrix}$ is :

$a^3+b^3+c^3-3abc$

The correct answer is Option (3) → $a^3+b^3+c^3-3abc$

$Δ=\begin{vmatrix} a-b & b+c & a\\ b-c & c+a & b \\ c-a & a+ b & c\end {vmatrix}$

$C_1→C_1+C_2-C_3$

$⇒Δ=\begin{vmatrix} c & b+c & a\\ a & a+c & b \\ b & a+ b & c\end {vmatrix}$

$C_2→C_2-C_1$

$Δ=\begin{vmatrix} c & b & a\\ a & c & b \\ b & a & c\end {vmatrix}$

$C_1→C_1+C_2+C_3$

$Δ=\begin{vmatrix} a+b+c & b & a\\ a+b+c & c & b \\ a+b+c & a & c\end {vmatrix}$

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

$R_3→R_3-R_2$

$R_2→R_2-R_1$

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

$Δ=(a+b+c)(b^2+c^2-2bc+a^2-ac-ab+bc)$

$=a^3+b^3+c^3-3abc$