The matrix integro-differential equation \((1)\quad X'(t)=X(t)B(t)+\int^{t}_{0}X(s)K(t,s)ds+F(t),\) is considered, where B(t) and K(t,s) are \(n\times n\) matrices definite and continuous on \(0\leq t<\infty\) and \(0\leq s\leq t<\infty,\) X(t) and F(t) are \(n\times n\) matrices with F(t) continuous on \(0\leq t<\infty\). The asymptotic behaviour of solutions to system (1) with matrix B(t) which is not necessarily stable is analysed. Estimates are obtained via the method of integral inequalities.