Question

evaluate by properties of determinants​

Answer 1

Answer:

Let the determinant be denoted by Δ then

Δ= $\left[\begin{array}{ccc}1&a&a^{2} \\1&b&b^{2} \\1&c&c^{2} \end{array}\right]$ + $\left[\begin{array}{ccc}1&a&-bc\\1&b&-ca\\1&c&-ab\end{array}\right]$

Multiply the rows of 2nd determinant by a, b & respectively and divide the determinant by abc.

Δ= $\left[\begin{array}{ccc}1&a&a^{2} \\1&b&b^{2} \\1&c&c^{2} \end{array}\right] -\frac{1}{abc} \left[\begin{array}{ccc}a&a^{2} &abc\\b&b^{2} &abc\\c&c^{2} &abc\end{array}\right]$

Δ= $\left[\begin{array}{ccc}1&a&a^{2} \\1&b&b^{2} \\1&c&c^{2} \end{array}\right] - \left[\begin{array}{ccc}a&a^{2} &1\\b&b^{2} &1\\c&c^{2} &1\end{array}\right]$