Some new oscillation criteria were established for the second order nonlinear matrix differential system $ (a(t)\X’(t))’+b(t)\X’(t)+\Q(t)f(\X(t))= 0,t\geqslant t_ 0 0,$ where $\Q(t),$ $f’(\X(t))$ are $n \times n$ matrices with $f’(\X(t))$ positive definite, and $a(t),$ $b(t)$ are real-valued functions. The criteria were presented in the form of $\lim \sup\lambda_ 1 const $ by using a particular function $\phi(t,s,r)$ defined as $\phi(t,s,r)=(t-s)^ \alpha (s-r)^ \beta $, where $\alpha,\ \beta \frac 1 2 $ are constants and $r \geqslant t_0.$ Our results improve many known oscillation results.