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## Find all solutions to the systems of equations by row-reducing (1)

Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.1

Solution: The matrix of coefficients is
$$\left[\begin{array}{cc}1-i&-i\\2&1-i\end{array}\right].$$Row reducing
$$\rightarrow \left[\begin{array}{cc}2&1-i\\1-i&-i\end{array}\right]\rightarrow\left[\begin{array}{cc}2&1-i\\0&0\end{array}\right]$$Thus $2x_1+(1-i)x_2=0$. Thus for any $x_2\in\mathbb C$, $(\frac12(i-1)x_2,x_2)$ is a solution and these are all solutions.