Hello!

I'm working on a program using Wolfram Mathematica that can give us the exponential matrix as a result. I have a 3x3 system as a base (but it may work with a nxn one), which gives me 3 linear equations that involve alphas and lambdas with their respective "t's". So far, I don't know how to turn that system into a matrix to get this form: e^( λi) = A* α . I would like to know if there is a way where I can take those α's away from my equations (attached image) and put them into an independent vector, so I can proceed to to this: α = A^-1 * e^( λi); "A" would be the matrix formed from the Eigenvalues and alphas polynomial equation showed below.

I would appreciate if you could help me with this situation.

Greetings!

P.S. I'm not allowed to use any shortcut like Solve or Matrix Exp.

Thank you!