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  4. The value of the determinant | 1&a&a2&-bc 1&b&b2&-ca 1 &c&c2&-ab |

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Q. The value of the determinant $ \begin{vmatrix} 1&a&a^2&-bc\\ 1&b&b^2&-ca\\ 1 &c&c^2&-ab \end{vmatrix}$

Determinants

Solution:

$\begin{vmatrix}1 &a&a^{2}-bc\\ 1 &b&b^{2}-ca\\ 1 &c&c^{2}-ab\end{vmatrix} = \begin{vmatrix}1 &a&a^{2}\\ 1 &b&b^{2}\\ 1 &c&c^{2}\end{vmatrix} - \begin{vmatrix}1 &a&bc\\ 1 &b&ca\\ 1 &c&ab\end{vmatrix}$
= $\begin{vmatrix}1 &a&a^{2}\\ 1 &b&b^{2}\\ 1 &c&c^{2}\end{vmatrix} - \frac{1}{abc} \begin{vmatrix}a &a^{2}& abc\\ b&b^{2}& abc\\ c&c^{2}&abc\end{vmatrix} $
= $\begin{vmatrix}1 &a&a^{2}\\ 1 &b&b^{2}\\ 1 &c&c^{2}\end{vmatrix} - \frac{abc}{abc} \begin{vmatrix} a &a^2&1\\b& b^{2}& 1\\ c& c^{2}&1\end{vmatrix} $
= $\begin{vmatrix}1 &a&a^{2}\\ 1 &b&b^{2}\\ 1 &c&c^{2}\end{vmatrix} - \begin{vmatrix}1 &a&a^{2}\\ 1 &b&b^{2}\\ 1 &c&c^{2}\end{vmatrix} = 0$