Practicing Success
The value of the determinant $\begin{vmatrix}b+c &a-b &a\\c+a &b-c &b\\a+b &c-a &c\end{vmatrix}$, is |
$a^3 +b^3 + c^3 - 3abc$ $3abc-a^3-b^3-c^3$ $3 abc + a^3 +b^3 + c^3$ none of these |
$3abc-a^3-b^3-c^3$ |
We have, $\begin{vmatrix}b+c &a-b &a\\c+a &b-c &b\\a+b &c-a &c\end{vmatrix}$ $=\begin{vmatrix}a+b+c&-b &a\\b+c+a &-c &b\\c+a+b &-a &c\end{vmatrix}$ [Applying $C_1→C_1 + C_3; C_2 →-C_2-C_3$] $=-(a+b+c)\begin{vmatrix}1&b &a\\1&c &b\\1&a &c\end{vmatrix}$ $=-(a+b+c) \begin{vmatrix}1&b &a\\0&c-b &b-a\\0&a-b &c-a\end{vmatrix}$ [Applying $=-(a+b+c)(a^2 + b^2 + c^2 -ab-bc - ca)$ $=-(a^3+b^3+c^3-3abc)$ |