Given the following matrix
$$A = \left[
\begin{array}{ccc}
-1-3i & -8-10i & 0-3i \\
-7-3i & -4-9i & -3-2i \\
11-3i & -16-12i & 6-5i
\end{array}
\right] $$
Find the general solution x to Ax = 0.
I am confused by what the "general" solution means; does anyone have any hints as to how to start solving this problem?
Best Answer
A general solution is a set of all possible solutions to the given equation. Here we need to perform Gaussian elimination to get a row-reduced form of $A$; I got (using SymPy) $$\begin{bmatrix} 1&0&\frac1{7730}(4001-597i)\\ 0&1&\frac1{3865}(218+464i)\\ 0&0&0\end{bmatrix}$$ The general solution remains the same with this reduced $A$, so it is $$(x,y,z)=\left\{\left(-\frac1{7730}(4001-597i)z,-\frac1{3865}(218+464i)z,z\right):z\in\mathbb C\right\}$$