Unit 5 Section 3 - 5.3 Matrices and Determinants...

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5.3 Matrices and Determinants Objectives: After completing this section, you should be able to solve systems of linear equations using the following methods: 1. Matrices and Gaussian elimination 2. Matrices and Gauss-Jordan elimination 3. Cramer’s rule In Sections 1 and 2 we learned how to solve systems of linear equations in two and three variables using elimination by addition and its variant, Gaussian elimination. We can organize more efficiently our work for solving linear systems using these methods, and two other methods you will see later, by introducing a mathematical form called a matrix . Corresponding to a linear system of equations is an augmented matrix which contains all of the information in the system without the x, y , and z variable labels to carry around. In this section we will revisit Gaussian elimination but this time using augmented matrices. Keep This in Mind A matrix is a rectangular array of numbers written within brackets that contains rows and columns. If m and n are positive integers, an m x n (read “ m by n” ) matrix is a rectangular array of numbers mn m m n n a a a a a a a a a 2 1 2 22 21 1 12 11 with m rows and n columns in which each number ij a is referred to as an entry of the matrix in the i th row and j th column. The expression m x n is called the size or dimension of the matrix. A matrix with n rows and n columns is called a square matrix of order n. A matrix with only one is called a column matrix , and a matrix with only one row is called a row matrix . Row 1 Row 2 Row m Col. 1 Col. 2 Col. n
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Unit 5. Systems of Equations and Inequalities Section 3 page 2 _____________________________________________________________________________________ Example 5.3.1 Below are two matrices: 7 0 2 4 3 1 A 2 5 0 3 8 6 1 0 15 1 0 4 B We usually name a matrix by a single capital letter, such as A , B, C, and so on. Matrix A above has six entries arranged in two rows and three columns, while matrix B has 12 entries arranged in three rows and four columns. So we call A as a 2 x 3 matrix and B as a 3 x 4 matrix. In matrix A we have 7 23 a and 3 12 a , while in matrix B we have 6 23 b and . 3 31 b Matrices Associated with Systems of Linear Equations Consider the following system of linear equations in the workstation problem of Example 5.1.2 in Section1: . 16 38 3 2 y x y x Related to this system are the following matrices: 1 1 3 2 16 38 16 1 1 38 3 2 The augmented matrix is what we need for Gaussian elimination. It is called “ augmented ” because it includes the right-hand side constants of the equations. The other matrices will be used later in this section. The augmented matrix contains the essential parts of the linear system – both the coefficients and
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Unit 5 Section 3 - 5.3 Matrices and Determinants...

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