Matrix Calculation Error: Finding the Correct Solution

[SoggyBottoms]

Homework Statement

See attached image.

The Attempt at a Solution

I get a different solution: First multiply by $\sqrt{2}$, then ${1 \choose -1} = {c + d \choose ci - di}$. So we get $c + d = 1$ and so $(1 - d)i - di = -1$. Solving the last one gives $2di = 1 + i$, so $d = \frac{1 + i}{2i} = \frac{1 + i}{2i}\frac{i}{i} = \frac{i - 1}{-2} = \frac{1 - i}{2}$. But that's the answer for c. Where does my solution go wrong?

 

[NascentOxygen]

The book is wrong. You can see this by substituting their "solution" into the original equation.

 

[SoggyBottoms]

Thanks.