m = 
      Table[ Random[Integer, {0, 100}], {i, 1, 5}, {j, 1, 
          5}]

Inverse[m]

v  =  
      Table[Random[Integer, {0, 100}], {i, 1, 5}]

Solve[ 
      m .  {a, b, c, d, e}  ==  v,  {a, b, c, d, e}]

Inverse[m]  .  v

N[Eigenvalues[m]]

N[Eigenvectors[m]]

Det[m]