Complex Eigenvalues of system of DE

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Homework Statement
Please see below.
Relevant Equations
Please see below
For this problem,
1715292619315.png

1715292474940.png

Can someone please explain to me how they got from the orange step to the yellow step?

I am confused how the two expressions are equivalent.

Thanks!
 

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Just perform the multiplications and collect real and imaginary terms in their own matrices.
 
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ChiralSuperfields said:
I am confused how the two expressions are equivalent.
The yellow expression contains a sign error; the second term should be:$$
i\left(\begin{array}{c}
-e^{-2t}\sin3t\\
+e^{-2t}\cos3t
\end{array}\right)
$$
 
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Thank you for your replies @Orodruin and @renormalize ! Yeah that was part of my confusion is that there is a typo
1715294433735.png

It makes sense now that I know it is a typo!

Thanks!
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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