The author considers the hyperbolic system \[ \begin{gathered} [I_4 D_t+ A(t) D_x+ B(t)]u(t,x)= 0,\\ u(0,x)= u_0(x)\end{gathered} \] in \(\Omega= [0,T]\times \mathbb{R}^1_x\) where \(I_4\) denotes the unit matrix of order 4 and \[ A(t)= \begin{pmatrix} \lambda(t) & 1 & 0 & 0\\ 0 &\lambda(t) & a(t) & 0\\ 0 & 0 & \mu(t) & 1\\ 0 & 0 & 0 & \mu(t)\end{pmatrix}, \] \(\lambda(t)\), \(\mu(t)\), \(a(t)\) are real smooth functions, with some assumptions. The author determines completely the Gevrey indices for the well-posedness of the Cauchy problem; this proves that the maximal multiplicity of the zeros of the minimal polynomial of the principal part does not give, in general, the appropriate index for the Gevrey well posedness.