Eigenvalues, eigenvectors, and time evolution.

A. Given matrix A = 1 2 0 where i found the eigenvalues to be 3hw, 0, -3hw.
2 0 2
0 2 -2

then if at t=0 state-vector is 1 THEN compute the state vector at time t.
0
0

Please explain me how to get this, Im very confused. I know its some sort of matrix multiplication but why?

b. If A = 1 0 0 is an observable. FIND its expectation at time t and compute the
0 2 0
0 0 0

probability that measuring A at time t gives the result 1.

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