Complex number in a 3x3 matrix

Question

To calculate this should I multiply conjugate of complex number? If I should I have no idea how to start...

Thanks

Answer 1

Just use your usual methods of solving linear systems of equations, but using arithmetic of complex numbers.

Answer 2

You can calculate the inverse of $M= \begin{bmatrix} 2 & 1 & 1 \\ 0 & 2 & 1 \\ 1 & i & 1 \end{bmatrix}$.

$M^{-1} = \begin{bmatrix} (8+i)/13 & (-5+i)/13 &(-3-2i)/13 \\ (3+2i)/13 &(3+2i)/13 &(-6-4i)/13 \\ (-6-4i)/13 &(7-4i)/13 &(12+8i)/13 \end{bmatrix}$.

And the solution is $M^{-1} \begin{bmatrix} 3+3i \\ 5+i \\ 1+4i \end{bmatrix} = \begin{bmatrix} x = i \\ y = 2 \\ z = 1+i\end{bmatrix} $.