Some explicit formulas for the matrix exponential

Formulas are derived for the exponential of an arbitrary 2×2 matrix in terms of either its eigenvalues or entries. These results are then applied to the second-order mechanical vibration equation with weak or strong damping. Some formulas for the exponential of n×n matrices are given for matrices that satisfy an arbitrary quadratic polynomial. Besides the above-mentioned 2×2 matrices, these results encompass involutory, rank 1, and idempotent matrices. Consideration is then given to n×n matrices that satisfy a special cubic polynomial. These results are applied to the case of a 3×3 skew symmetric matrix whose exponential represents the constant rotation of a rigid body about a fixed axis