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

MATH152 April 2022

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Question A16
Solve the system of equations corresponding to the augmented matrix $$\left[{\begin{array}{cc|c}1&i&2\\-i&2&1\end{array}}\right].$$ You may assume that the unknown variables are ${\textstyle z_{1}}$, ${\textstyle z_{2}}$. Simplify your final answers ${\textstyle z_{1}}$, ${\textstyle z_{2}}$ to the form ${\textstyle a+bi}$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall how complex multiplication works.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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 ${\textstyle z_{2}=1+2i}$.