In this paper，we first consider a matrix analogue of H?lder's inequality. Furthermore，we obtain a matrix analogue of the arithmetric-geometric-mean inequality. Then we give an affirmative answer to two open questions of R.Bellman.^[1] After that，we obtain a matrix analogue of Cauchy-Schwarz inequality and its complementary (or reverse) inequalities. We derive a similar result for nonnegative matrices.