Science:Math Exam Resources/Courses/MATH152/April 2022/Question A16/Solution 1

The tricky part of this question is the multiplication of complex numbers, otherwise it’s a straightforward system of equations! Above the arrows below, we use the notation $L_{i,j}(k)$ for the row-operation "add $k$ times row $j$ to row $i$."

$$\left[{\begin{array}{cc|c}1&i&2\\-i&2&1\end{array}}\right]\xrightarrow {L_{2,1}(i)} \left[{\begin{array}{cc|c}1&i&2\\0&1&1+2i\end{array}}\right]\xrightarrow {L_{1,2}(-i)} \left[{\begin{array}{cc|c}1&0&2-i(1+2i)\\0&1&1+2i\end{array}}\right]$$

We have: ${\textstyle z_{1}=2-i(1+2i)=4-i}$ and ${\textstyle z_{2}=1+2i}$.