Fourier transform of a matrix!

Question

I have a non-trivial task of computing the Fourier transform of a matrix $\text{mat}[\alpha,\beta, t]$ such that the Fourier transform looks like $\text{matFT}[\alpha,\beta, \omega]$. With $\text{mat}[\alpha,\beta, t]$ given by

 mat[α_, β_, t_] = {
  {(Sqrt[α^2 - 4 β^2] Cosh[1/4 t Sqrt[α^2 - 4 β^2]] - α Sinh[1/4 t Sqrt[α^2 - 4 β^2]])^2/(-4 β^2 + α^2 Cosh[1/2 t Sqrt[α^2 - 4 β^2]] - α Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]]),
   (I β (-α + α Cosh[1/2 t Sqrt[α^2 - 4 β^2]] - Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]]))/(4 β^2 - α^2 Cosh[1/2 t Sqrt[α^2 - 4 β^2]] + α Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]])
  },

  {-((I β (-α + α Cosh[1/2 t Sqrt[α^2 - 4 β^2]] - Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]]))/(4 β^2 - α^2 Cosh[1/2 t Sqrt[α^2 - 4 β^2]] + α Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]])),
   -((4 β^2 Sinh[1/4 t Sqrt[α^2 - 4 β^2]]^2)/(4 β^2 - α^2 Cosh[1/2 t Sqrt[α^2 - 4 β^2]] + α Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]]))
  }
};

How can this be achieved in Mathematica?