The matrix A is almost a rotation matrix. I suspect you want to find the Euler angles. There are various conventions for these, but one of them is
You won't get that exactly, but we can try to get a good fit using LSSolve in the Optimization package. There's a slight complication: gamma is not a good name for a variable in LSSolve, because gamma is actually the name of a constant in Maple.
After entering the matrix above as B and your matrix as A:
> Optimization[LSSolve](subs(gamma=c, [seq(seq(A[i,j]-B[i,j],i=1..3),j=1..3)]),alpha=0..2*Pi,beta=0..2*Pi,c=0..2*Pi);
The first number being so small indicates an extremely good fit. Here's the matrix with these values of the angles.
> subs(gamma=c, %, B);