The concepts of polynomials and matrices essentially expand and enhance the elementary arithmetic of numbers. Once introduced, polynomials and matrices open up new interesting mathematical challenges which extend to new fields of mathematical explorations within university mathematics. We present an aspect of a rather elementary exploration of polynomials and matrices, which offers a new perspective and an interesting matrix analogue to the concept of a zero of a polynomial. The discussion offers an opportunity for better comprehension of the fundamental concepts of polynomials and matrices. As an application we are led to the meaningful questions of linear algebra and to an easy proof of the otherwise advanced and abstract Cayley-Hamilton theorem.

The ellipse is well known to students of mathematics at an advanced high school level and at an undergraduate level through its associations with, say, conic sections and their deployment as hardware in dual reflector antenna systems. Perhaps less well known is its appearance on the software side, so to speak, for example in RF (Radio Frequency) engineering in the context of polarisation, where it is known as the polarisation ellipse. It is the purpose here to discuss this manifestation and dwell particularly on the concept of a circumscribing rectangle for this ellipse because such aspects are of interest to the RF engineer who may be involved with measurements to determine its properties. This exercise affords an interesting insight into a very practical application of tertiary level mathematics and whilst it is of course helpful to appreciate the physical concepts involved, a lack of such knowledge does not detract from an appreciation of the underlying sums, which should be within the grasp of the above, interested student.

This problem was posed in the 1930s by Paul Erdős [1]. It is the simplest case of a still unsolved conjecture.

If k2 + 1 squares with no interior point in common are drawn inside a unit square then the maximum possible sum of their side-lengths is k [2].

We shall use the notation S(n) to denote the maximum possible sum of the side-lengths for n squares drawn with no interior point in common inside a unit square. The main aim of this article will be to develop an approach to the study of the function S which will give surprisingly simple proofs of a number of known results. This approach will then be used to prove a new result about the asymptotic behaviour of S.