How would one solve a complex equation system solely using a cartesian representation of complex numbers by hand? For instance, take the following linear equation system:
$(1+i)z_1 – z_2 = i$
$(1-i)z_1 + (1+i)z_2 = 1$
This system contains both complex variables and complex coefficients. Is it even possible to solve an equation system like this? How would one go about it? Does it have multiple solutions since it includes complex numbers?
Best Answer
Matrices with complex entries can be solved just as matrices with real entries.
$$ \left[ \begin{array}{cc|c} 1+i&-1&i\\ 1-i&1+i&1 \end{array} \right] $$
Reduce the matrix to RREF, using the exact same methods as real matrices. The only difference will be that you may have to do a bit of complex arithmetic.
In this case, you should end up with something like:
$$ \left[ \begin{array}{cc|c} 1&0&i/(i+1)\\ 0&1&0 \end{array} \right] $$