Question

what is the value of $\left|\begin{array}{c}a{a}^2{a}^3-1\\ b{b}^2{b}^3-1\\ c{c}^2{c}^3-1\end{array}\right|=?$

Answer 1

L.H.S

$\Rightarrow \left|\begin{array}{ccc}a& a^2& a^3-1\\ b& b^2& b^3-1\\ c& c^2& c^3-1\end{array}\right|$

Apply, $C_2\to C_2-C_1$

$\Rightarrow \left|\begin{array}{ccc}a& a^2-a& a^3-1\\ b& b^2-b& b^3-1\\ c& c^2-c& c^3-1\end{array}\right|$

On taking common $(a-1)(b-1)(c-1)$, we get

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

Apply, $C_1\to C_1-C_2$

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

$\Rightarrow 0$.

R.H.S

Hence, proved.