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Abstract The purpose of this work is to obtain several Lyapunov inequalities for the nonlinear dynamic systems { x Δ ( t ) = − A ( t ) x ( σ ( t ) ) − B ( t ) y ( t ) | B ( t ) y ( t ) | p − 2 , y Δ ( t ) = C ( t ) x ( σ ( t ) ) | x ( σ ( t ) ) | q − 2 + A T ( t ) y ( t ) , $$\left \{ \textstyle\begin{array}{l} x^{\Delta}(t)= -A(t)x(\sigma(t))-B(t)y(t)|\sqrt{B(t)}y(t)|^{p-2}, \\ y^{\Delta}(t)= C(t)x(\sigma(t))|x(\sigma(t))|^{q-2}+A^{T}(t)y(t), \end{array}\displaystyle \right . $$ on a given time scale interval [ a , b ] T $[a,b]_{\mathbb{T}}$ ( a , b ∈ T $a,b\in{\mathbb{T}}$ with σ ( a ) < b $\sigma(a)< b$ ), where p , q ∈ ( 1 , + ∞ ) $p,q\in (1,+\infty)$ satisfy 1 / p + 1 / q = 1 $1/p+1/q=1$ , A ( t ) $A(t)$ is a real n × n $n\times n$ matrix-valued function on [ a , b ] T $[a,b]_{\mathbb{T}}$ such that I + μ ( t ) A ( t ) $I+\mu(t)A(t)$ is invertible, B ( t ) $B(t)$ and C ( t ) $C(t)$ are two real n × n $n\times n$ symmetric matrix-valued functions on [ a , b ] T $[a,b]_{ \mathbb{T}}$ , B ( t ) $B(t)$ is positive definite, and x ( t ) $x(t)$ , y ( t ) $y(t)$ are two real n-dimensional vector-valued functions on [ a , b ] T $[a,b]_{\mathbb{T}}$ .