{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MM1_Chapter_10

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern