Why can I not get this matrix to make any sense?

I am trying to solve this differential equation system:

I'm trying to write this into a matrix in the form of:

x' = A x

Where "A" is my matrix for this equation system. But when I try to create the matrix out of my own logic thinking, I get this:

This doesn't make any sense though since the determinant here is 0, which it shouldn't be. So what am I doing wrong? :(

Re: Why can I not get this matrix to make any sense?

Why do you say that does not make sense? Why can the determinant not be 0? That matrix has eigenvalues 0 (because the determinant is 0), i and -i. That means each will be of the form A+ B cos(x)+ C sin(x).

But, personally, I wouldn't use a matrix solution. Differentiating the first equation again, we have so that . That equation has general solution . Then so that . Finally, so that just as I said.

Re: Why can I not get this matrix to make any sense?

Re: Why can I not get this matrix to make any sense?

Quote:

Originally Posted by

**HallsofIvy** Why do you say that does not make sense? Why can the determinant not be 0? That matrix has eigenvalues 0 (because the determinant is 0), i and -i. That means each

will be of the form A+ B cos(x)+ C sin(x).

But, personally, I wouldn't use a matrix solution. Differentiating the first equation again, we have

so that

. That equation has general solution

. Then

so that

. Finally,

so that

just as I said.

How did you conclude that the matrix has the eigenvalues i and -i?

My assignment says I need to find the complete complex solution for the differential equation system. And that the facit is this:

By looking at the facit, it says that the eigenvalues are 0, i and -i but our textbook didn't mention anything about a matrix with eigenvalues of 0 so I have no idea how to conclude this.

We haven't been taught any other way of solving this. According to my textbook, once I have the eigenvalues and eigenvectors, I can insert them into this solution:

where "lambda" is my eigenvalue and "v" is my eigenvectors. Since I have a system of 3 equations, I should end up with 3 solutions.

But nobody ever mentioned to me that a matrix with determinant 0 also has the eigenvalues i and -i? :(

I understand that -i is the conjugated value of i.