M1410801 - (Section 8.1 Matrices and Determinants 8.01...

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(Section 8.1: Matrices and Determinants) 8.01 CHAPTER 8: MATRICES and DETERMINANTS The material in this chapter will be covered in your Linear Algebra class ( Math 254 at Mesa ). SECTION 8.1: MATRICES and SYSTEMS OF EQUATIONS PART A: MATRICES A matrix is basically an organized box (or “array”) of numbers (or other expressions). In this chapter, we will typically assume that our matrices contain only numbers. Example Here is a matrix of size 2 × 3 (“2 by 3”), because it has 2 rows and 3 columns: 1 0 2 0 1 5 The matrix consists of 6 entries or elements . In general, an m × n matrix has m rows and n columns and has mn entries. Example Here is a matrix of size 2 × 2 (an order 2 square matrix): 4 1 3 2 The boldfaced entries lie on the main diagonal of the matrix. (The other diagonal is the skew diagonal.)
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(Section 8.1: Matrices and Determinants) 8.02 PART B: THE AUGMENTED MATRIX FOR A SYSTEM OF LINEAR EQUATIONS Example Write the augmented matrix for the system: 3 x + 2 y + z = 0 2 x z = 3 Solution Preliminaries: Make sure that the equations are in (what we refer to now as) standard form, meaning that … • All of the variable terms are on the left side (with x , y , and z ordered alphabetically), and • There is only one constant term, and it is on the right side. Line up like terms vertically. Here, we will rewrite the system as follows: 3 x + 2 y + z = 0 2 x z = 3 (Optional) Insert “1”s and “0”s to clarify coefficients. 3 x + 2 y + 1 z = 0 2 x + 0 y 1 z = 3 Warning: Although this step is not necessary, people often mistake the coefficients on the z terms for “0”s.
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(Section 8.1: Matrices and Determinants) 8.03 Write the augmented matrix: Coefficients of Right x y z sides 3 2 1 2 0 1 0 3 Coefficient matrix Right-hand side (RHS) Augmented matrix   We may refer to the first three columns as the x -column, the y -column, and the z -column of the coefficient matrix. Warning: If you do not insert “1”s and “0”s, you may want to read the equations and fill out the matrix row by row in order to minimize the chance of errors. Otherwise, it may be faster to fill it out column by column. The augmented matrix is an efficient representation of a system of linear equations, although the names of the variables are hidden.
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(Section 8.1: Matrices and Determinants) 8.04 PART C: ELEMENTARY ROW OPERATIONS (EROs) Recall from Algebra I that equivalent equations have the same solution set. Example Solve: 2 x 1 = 5 2 x 1 = 5 2 x = 6 x = 3 Solution set is 3 { } . To solve the first equation, we write a sequence of equivalent equations until
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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M1410801 - (Section 8.1 Matrices and Determinants 8.01...

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