We consider the seond order differential system
(
1
)
Y
+
Q
(
t
)
Y
=
0
(1)\,Y + Q(t)Y = 0
, where
Q
Q
,
Y
Y
are
n
×
n
n \times n
matrices with
Q
=
Q
(
t
)
Q = Q(t)
a continuous symmetric matrix-valued function,
t
∈
[
a
,
+
∞
]
t \in [a,\, + \infty ]
. We obtain a number of sufficient conditions in order that all prepared solutions
Y
(
t
)
Y(t)
of
(
1
)
(1)
are oscillatory. Two approaches are considered, one based on Riccati techniques and the other on variational techniques, and involve assumptions on the behavior of the eigenvalues of
Q
(
t
)
Q(t)
(or of its integral). These results extend some well-known averaging techniques for scalar equations to system
(
1
)
(1)
.