His mathematical achievements were of the first rank. Finally, in 1863, he accepted the Sadlerian professorship in Cambridge and remained there for the rest of his life, valued for his administrative and teaching skills as well as for his scholarship. With no employment in mathematics in view, he took legal training and worked as a lawyer while continuing to do mathematics, publishing nearly 300 papers in fourteen years. This subject is quite old and was first studied systematically in 1858 by Arthur Cayley.Īrthur Cayley (1821-1895) showed his mathematical talent early and graduated from Cambridge in 1842 as senior wrangler. Furthermore, matrix algebra has many other applications, some of which will be explored in this chapter. These “matrix transformations” are an important tool in geometry and, in turn, the geometry provides a “picture” of the matrices. For example, the geometrical transformations obtained by rotating the euclidean plane about the origin can be viewed as multiplications by certain matrices. This “matrix algebra” is useful in ways that are quite different from the study of linear equations. While some of the motivation comes from linear equations, it turns out that matrices can be multiplied and added and so form an algebraic system somewhat analogous to the real numbers. In the present chapter we consider matrices for their own sake. Our aim was to reduce it to row-echelon form (using elementary row operations) and hence to write down all solutions to the system.
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