There is a linear equation, in matrix form (like a system) giving me a hard time, this is it:
|1, x, x^2, x^3 |
|x^3, 1, x, x^2 |
|x^2, x^3, 1, x | = 0
|x, x^2, x^3, 1 |
(hope it looks like a matrix)
1. Am I allowed to solve only one line, it sounds stupid but they look pretty the same, does it make any difference that there are four lines, (I am asking this cause I solved the first one and it gives three values for x: -1, i, -i)
2. But I am not satisfied with this, there must be some theoreme, some topic I am missing, is it anything related to digonalization, inversion, eigen values-vectors, characteristic polynomial equation, please can someone give any idea what algebra topics I should study for this?
3. If someone could give a solution I would apreciate it a lot in order to compare it with mine (if I ever find one), I am so curious to know what is the solution.
4. They called it an "equation", I suppose because there is only one variable, x. Since they call it an equation why it is presented like a system?
5. I know how to use gaussian elemination, cramer etc, is this of any help or I should study cubic non linear systems (for which I have no idea so far)?
Thank you all,