Question

Show that:

$\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ a& b& c\\ 1& 1& 1\end{array}\right|=-(a-b)(b-c)(c-a)$

Answer 1

Given $\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ a& b& c\\ 1& 1& 1\end{array}\right|$

$c_1\to c_1-c_2$

$c_2\to c_2-c_3$

$\left|\begin{array}{ccc}{a}^2-b^2& b^2-c^2& c^2\\ a-b& b-c& c\\ 0& 0& 1\end{array}\right|$

$=1\left[\right(a^2-b^2\left)\right(b-c)-(a-b\left)\right(b^2-c^2\left)\right]$

$=(a-b)(b-c)[a+b-b-c]$

$=(-1)(a-b)(b-c)(c-a)$. $\left[henceproved\right]$