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This chapter covers principles of matrices. After completing this chapter students should be able to: complete matrix operations; solve linear systems using Gauss-Jordan method; Solve linear systems using the matrix inverse method and complete application problems.

Chapter overview

In this chapter, you will learn to:

  1. Do matrix operations.
  2. Solve linear systems using the Gauss-Jordan method.
  3. Solve linear systems using the matrix inverse method.
  4. Do application problems.

Introduction to matrices

Section overview

In this section you will learn to:

  1. Add and subtract matrices.
  2. Multiply a matrix by a scalar.
  3. Multiply two matrices.

A matrix is a rectangular array of numbers. Matrices are useful in organizing and manipulating large amounts of data. In order to get some idea of what matrices are all about, we will look at the following example.

Fine Furniture Company makes chairs and tables at its San Jose, Hayward, and Oakland factories. The total production, in hundreds, from the three factories for the years 1994 and 1995 is listed in the table below.

1994 1995
Chairs Tables Chairs Tables
San Jose 30 18 36 20
Hayward 20 12 24 18
Oakland 16 10 20 12
  1. Represent the production for the years 1994 and 1995 as the matrices A and B.
  2. Find the difference in sales between the years 1994 and 1995.
  3. The company predicts that in the year 2000 the production at these factories will double that of the year 1994. What will the production be for the year 2000?
  1. The matrices are as follows: A = 30 18 20 12 16 10 size 12{A= left [ matrix { "30" {} # "18" {} ##"20" {} # "12" {} ## "16" {} # "10"{}} right ]} {} B = 36 20 24 18 20 12 size 12{B= left [ matrix { "36" {} # "20" {} ##"24" {} # "18" {} ## "20" {} # "12"{}} right ]} {}
  2. We are looking for the matrix B A size 12{B - A} {} . When two matrices have the same number of rows and columns, the matrices can be added or subtracted entry by entry. Therefore, we get

    B A = 36 30 20 18 24 20 18 12 20 16 12 10 = 6 2 4 6 4 2 size 12{B - A= left [ matrix { "36" - "30" {} # "20" - "18" {} ##"24" - "20" {} # "18" - "12" {} ## "20" - "16" {} # "12" - "10"{}} right ]= left [ matrix {6 {} # 2 {} ## 4 {} # 6 {} ##4 {} # 2{} } right ]} {}
  3. We would like a matrix that is twice the matrix of 1994, i.e., 2A size 12{2A} {} .

    Whenever a matrix is multiplied by a number, each entry is multiplied by the number.

    2A = 2 30 18 20 12 16 10 = 60 36 40 24 32 20 size 12{2A=2 left [ matrix { "30" {} # "18" {} ##"20" {} # "12" {} ## "16" {} # "10"{}} right ]= left [ matrix {"60" {} # "36" {} ## "40" {} # "24" {} ##"32" {} # "20"{} } right ]} {}
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Before we go any further, we need to familiarize ourselves with some terms that are associated with matrices. The numbers in a matrix are called the entries or the elements of a matrix. Whenever we talk about a matrix, we need to know the size or the dimension of the matrix. The dimension of a matrix is the number of rows and columns it has. When we say a matrix is a 3 by 4 matrix, we are saying that it has 3 rows and 4 columns. The rows are always mentioned first and the columns second. This means that a 3 × 4 size 12{3 times 4} {} matrix does not have the same dimension as a 4 × 3 size 12{4 times 3} {} matrix. A matrix that has the same number of rows as columns is called a square matrix . A matrix with all entries zero is called a zero matrix . A square matrix with 1's along the main diagonal and zeros everywhere else, is called an identity matrix . When a square matrix is multiplied by an identity matrix of same size, the matrix remains the same. A matrix with only one row is called a row matrix or a row vector , and a matrix with only one column is called a column matrix or a column vector . Two matrices are equal if they have the same size and the corresponding entries are equal.

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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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