Vectors, Matrices and Geometry

CHAPTER SIX: MATRIX AND DETERMINANT

CHAPTER SIX MATRIX AND DETERMINANT Matrices are introduced in the last chapter as a systematic way of presenting the components of m vectors of Rn so that we can keep track of certain calculations being carried out on them. The chief concern of such calculations is to evaluate the rank of a matrix and to select linearly independent row vectors. In this chapter matrices are treated as individual algebraic entities on their own right. Sums and products of matrices as well as their properties are studied in the first part of this chapter. A particularly interesting and useful result is the interpretation of elementary transformations on matrices as multiplications by elementary matrices . Determinants of order 2 and 3 are mentioned briefly in the earlier chapters; they will be studied in some detail in the second part of this chapter. In conclusion we also make some useful observations on determinants of higher order so that readers will be able to see how they can be defined and what properties they have in common with determinants of lower order. 6.1 Terminology We recall that for any two positive integers m and n, an m x nmatrix A is a rectangular array of mn real numbers: in m rows and n columns. For easy reference, we shall introduce the following specific terminology and notations. The ordered pair (m, n) is called the order of the matrix A. Each of the mn real numbers 243 Vectors1 Matrices and Geometry aij (i =1,2,··· ,m; j = 1,2,··· ,n) is an element of A. The integer i is the row index and the integer j is the column index of the element aij j they indicate the position of the element aij in the matrix A. A matrix is usually denoted by an italic capital letter and its elements by the lower case italics of the same letter. The elements of a matrix are usually enclosed by a pair of elongated parentheses. Two matrices A and B are equal if and only if they have the same order and the same elements. Thus for an m x n-matrix A and a p x q-matrix B, A = B if and only if (i) m = p and n = q, and (ii) aij =bij for all i and j. Each horizontal sequence of n numbers of an m x n-matrix of A is called a row of A. The i-th row of A is a vector ri(A) = [ail, ai2, ... ,ain] of the vector space Rn. As its elements are arranged horizontally in A, it is also a 1 x n-matrix or a row matrix: Similarly the j-th column of A is a vector Cj (A) =[alj, a2j, ... ,amj] of Rm and is also an m x I-matrix or a column matrix: Finally by converting rows of A into columns, we obtain the transpose At of A: which is an n x m-matrix. Therefore (At)t = A. If we call the elements all, a22, ... ,aii, ... of A the diagonal elements of A, then A has min (m, n) diagonal elements which are identical to the diagonal elements of At. 244 Matrix and Determinant EXERCISES 1. Construct the 3 x 3 matrix defined by aij = 0 aii = 3i . for i =1= j 2. Construct the 4 x 4 matrix such that aij = least common multiple of i and j . 3. Construct the 5 x 5 matrix such that aij = 1 aij = 0 if li - jl is even or zero otherwise. 4. Find real numbers x, y, z such that ( 64)=(X+y+Z 5 3 y + z 5. Find real numbers a, b, c and d if X+z) . x+y (: ~) = (c=1d 2;:n. 6.2 Scalar multiple and sum We now begin treating the set of all m x n-matrices as a set of algebraic entities for any fixed integers m and n. Infact we shall carry out certain algebraic operations on them in a rather similar way as we have done with vectors. The multiple of an m x n-matrix A by a real number (scalar) r is the m x n-matrix rA whose elements are raij (i = 1,2,··· ,mj j = 1,2,··· ,n): 245 Vectors} Matrices and Geometry Therefore if r = 1, then lA = A. If r = 0, then all elements of OA are zero. This particular matrix is called...