In many areas of mathematical analysis, we encounter systems of ordered sets of elements, which could be sets of numbers, functions, or even equations. Determinants and matrices provide an elegant and an efficient computational tool for handling such systems. We already used some of the basic properties of matrices and determinants when we discussed coordinate systems and tensors. In this chapter, we give a formal treatment of matrices and determinants and their applications to systems of linear equations.
4.1 BASIC DEFINITIONS
We define a rectangular matrix of dimension as an array 
where the elements of could be numbers, functions, or even other matrices. An alternate way to write a matrix is
where is called the th element or the component of . The first subscript denotes the row number, and the second subscript denotes the ...