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MM1_Chapter_10 - Chapter 10 Matrices In Section 10.1 we...

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Chapter 10 Matrices In Section 10.1 we introduce m × n matrices which are rectangular arrays of real numbers consisting of m rows and n columns. For example, A = p 2 1 3 1 3 1 P is a 2 × 3 matrix with two rows and three columns. Vectors which we have encoun- tered in Chapter 9, are a special case of matrices. In Section 10.2, we learn how to add and subtract matrices of the same size and how to multiply matrices with scalars . We also learn how to multiply matrices with each other. In Sec- tion 10.3, we introduce the determinant of a square matrix, which is a scalar. In Section 10.4, we learn how to compute the inverse matrix of a matrix with deter- minant di±erent from zero. Finally we will use matrices and their inverse matrices to solve linear systems of equations in Section 10.5. 10.1 Introduction We start by introducing matrices as rectangular arrays of real numbers. In the previous chapter we have already encountered vectors which are special cases of matrices. 263
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264 10. Matrices Defnition 10.1 (matrix) An m × n (read ‘ m by n ’) matrix A = ( a i,j ) = ( a i,j ) i =1 , 2 ,...,m j =1 , 2 ,...,n is a rectangular array of real numbers containing m rows and n columns A = a 1 , 1 a 1 , 2 a 1 , 3 ··· a 1 ,n a 2 , 1 a 2 , 2 a 2 , 3 a 2 ,n a 3 , 1 a 3 , 2 a 3 , 3 a 3 ,n . . . . . . . . . . . . . . . a m, 1 a m, 2 a m, 3 a m,n The entry in the intersection of the i th row and j th column is denoted by a i,j and is called the ( i, j ) th entry of the matrix. That is, the frst index i oF a i,j denotes the row and the second index j oF a i,j denotes the column . Example 10.2 (matrices) (a) The matrix A = ( a i,j ) = p 2 3 1 4 2 1 0 5 P has m = 2 rows and n = 4 columns, and hence is a 2 × 4 matrix. Here we have a 1 , 1 = 2, a 1 , 2 = 3, a 1 , 3 = 1, a 1 , 4 = 4, and a 2 , 1 = 2, a 2 , 2 = 1, a 2 , 3 = 0, a 2 , 4 = 5. (b) The matrix B = ( b i,j ) = 1 2 3 2 1 4 7 5 9 3 4 8 has m = 4 rows and n = 3 columns, and hence is a 4 × 3 matrix. (c) The row vector a = (1 , 4 , 5) has m = 1 rows and n = 3 columns, and hence is a 1 × 3 matrix. (d) The column vector b = 1 2 3 has m = 3 rows and n = 1 columns, and hence is a 3 × 1 matrix. a
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10. Matrices 265 Defnition 10.3 (equal matrices) Two matrices are said to be equal if they have the same size , that is, the same numbers of rows and the same number of columns, and if all corresponding entries are the same . In formulas, an m × n matrix A = ( a i,j ) and an × k matrix B = ( b i,j ) are equal if m = and n = k and a i,j = b i,j for all i = 1 , 2 , . . ., m and all j = 1 , 2 , . . ., n. If two matrices A and B are equal we may write A = B , and if two matrices are not equal we may write A n = B . Example 10.4 (equal and unequal matrices) (a) Consider the matrices A = p 2 3 1 4 2 1 0 5 P and B = p 2 3 1 2 1 0 P . The matrix A is a 2 × 4 matrix, and B is an 2 × 3 matrix. Since the matrices have not the same dimension, we have A n = B .
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