Abstract: We shall show some easy matrix techniques to come up with interesting results like the maximum modulus principle and the von Neumann’s inequality. This involves forming polynomials of matrices. So, we shall talk about the functions of matrices. Suppose T is an n by n matrix with the largest singular value not larger than 1. The von Neumann’s inequality is a fundamental result which states that for a polynomial p and a matrix T as above, the largest singular value of p(T) is not larger than 1. Interestingly, this has a relation with complex analysis. The method of proof of von Neumann’s inequality produces a new proof of the maximum modulus principle as well.