Clear[v, p, e]
A = {{1, 0, 0}, {v, p, 0}, {v^2/2 + e, p*v, 
          p}}
B = {{v, p, 0}, {v^2 + (g - 1)*e, 
          2  p*v, (g - 1)*p}, {(v^3)/2 + (g - 1)*e*v + 
            e*v, (3/2)*p*v^2 + p*e + (g - 1) p*e, (g - 1)*p*v + 
            p*v}}
MatrixForm[Simplify[Inverse[A] . B]]
Eigensystem[Inverse[A] . B]