Hello! I'm trying to solve
$$\left|\begin{matrix} x^3-1 & x^2-1 & x – 1\\ x^3-8 & x^2-4 & x – 2 \\ x^3-27 & x^2-9 & x – 3 \end{matrix}\right| = 0.$$
(see here). Essentially, it is a determinant of a 3×3 matrix that is equal to zero.
I think I need to take advantage of the properties of a zero determinant (has infinite solutions) and replace x with something else…. That way, the system gets way less complex and I will be able to do Gauss elimination and what not… I've tried though changing x to a bunch of things (2x, x^2, √x etc) but i cant seem to find a pattern.
Am I missing something? Any help would be greatly appreciated ^^
Best Answer
Obviously $x = 1$, $x = 2$ and $x = 3$ are solutions. Moreover, by subtracting the first row from the second and the third, we arrive at the equation $$\left|\begin{matrix} x^3-1 & x^2-1 & x - 1\\ -7 & -3 & -1 \\ -26 & -8 & -2 \end{matrix}\right| = 0.$$ This determinant is now evidently a polynomial of degree $3$. As we already found three roots, these are all solutions to the equation.