<s>[INST] Find the determinant without row expansion Show that the determinant of the matrix 
\begin{bmatrix} 1& a& a^3\\
1& b& b^3\\  
1& c& c^3\end{bmatrix}
is $(a-b)(b-c)(c-a)(a+b+c)$ without expanding.
I was able to get out $(a-b)(b-c)(c-a)$ but couldn't complete.
 [/INST] \begin{bmatrix} 1& a& a^2&a^3\\
1& b& b^2 &b^3\\  
1& c& c^2 &c^3\\
1& X& X^2 &X^3
\end{bmatrix}is the well known Vandermonde determinant. When expend with respect to the last line, this is a polynomial
whose $X^2$ coefficient is the opposite of the result.
Now, expending 
$$
(c-a)(b-a)(c-b)(X-c)(X-b)(X-a) \\= (c-a)(b-a)(c-b)(X^3 - (a+b+c)X+\cdots)
$$gives the result $$
(a+b+c)(c-a)(b-a)(c-b)
$$
</s>