Book cover for Precalculus with Limits

Precalculus with Limits

Ron Larson

ISBN #9781439049099

2nd Edition

7,319 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces matrices as rectangular arrays of numbers that effectively represent systems of linear equations. It discusses how to determine the order of a matrix, form augmented and coefficient matrices, and perform elementary row operations. Key procedures such as Gaussian elimination (with back-substitution) and Gauss-Jordan elimination (leading to the reduced row-echelon form) simplify solving these systems. Mastering these techniques is essential for modeling and solving real-world problems in engineering, finance, and other fields.

Learning Objectives

1

Identify and write matrices including determining their order, and recognize special matrices like row and column matrices.

2

Define key terms including augmented matrix, coefficient matrix, and understand how they represent systems of linear equations.

3

Perform and document elementary row operations to transform matrices into row?echelon or reduced row?echelon forms.

4

Apply Gaussian elimination and Gauss-Jordan elimination techniques to solve systems of linear equations.

5

Use matrix operations to model and solve real-life problems such as electrical circuits, projectile motion, or financial applications.

Key Concepts

CONCEPT

DEFINITION

Matrix

A rectangular array of real numbers arranged in rows and columns, used to represent data or systems such as linear equations.

Order of a Matrix

The dimensions of a matrix given by the number of rows (m) and columns (n); written as an m × n matrix.

Square Matrix

A matrix with the same number of rows and columns (m = n).

Row Matrix

A matrix that contains only one row.

Column Matrix

A matrix that contains only one column.

Augmented Matrix

A matrix that includes the coefficient matrix of a system of linear equations along with an extra column for the constant terms.

Coefficient Matrix

A matrix composed only of the coefficients from a system of linear equations, excluding the constants.

Elementary Row Operations

Operations performed on the rows of a matrix including row interchange, scaling a row by a nonzero constant, and adding a multiple of one row to another; used to simplify matrices without changing the solution of the associated linear system.

Row-Echelon Form

A form of a matrix where all nonzero rows are above any rows of zeros, and the leading coefficient (first nonzero number from the left) of a nonzero row is always strictly to the right of the leading coefficient of the row above it.

Reduced Row-Echelon Form

A refined row-echelon form in which every leading 1 is the only nonzero entry in its column.

Gaussian Elimination

A method for solving systems of linear equations by transforming the corresponding augmented matrix into a row-echelon form using elementary row operations, followed by back-substitution.

Gauss-Jordan Elimination

An extension of Gaussian elimination that continues the elimination process to obtain a reduced row-echelon form, often eliminating the need for back-substitution.

Example Problems

Example 1

A rectangular array of real numbers that can be used to solve a system of linear equations is called a ________.

Example 2

A matrix is ________ if the number of rows equals the number of columns.

Example 3

For a square matrix, the entries $a_{11}$, $a_{22}$, $a_{33}$, $\ldots$, $a_{nn}$ are the ________ ________ entries.

Example 4

A matrix with only one row is called a ________ matrix, and a matrix with only one column is called a ________ matrix.

Example 5

The matrix derived from a system of linear equations is called the ________ matrix of the system.
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Step-by-Step Explanations

QUESTION

Given a matrix with m rows and n columns, how do you determine its order?

STEP-BY-STEP ANSWER:

Step 1: Count the number of rows in the matrix.
Step 2: Count the number of columns in the matrix.
Step 3: Express the matrix order as m × n.
Final Answer: The order of the matrix is m × n.

Determining the Order of a Matrix

QUESTION

How do you create the augmented matrix from a system of linear equations?

STEP-BY-STEP ANSWER:

Step 1: Write each equation in standard form, aligning like variables.
Step 2: Extract the coefficients and place them in order to form the coefficient matrix.
Step 3: Add a column on the right with the constants from each equation, using zeros where a variable is missing.
Final Answer: The resulting augmented matrix represents both the coefficients and the constant terms of the system.

Forming an Augmented Matrix

QUESTION

How can you use elementary row operations to convert a matrix into row-echelon form?

STEP-BY-STEP ANSWER:

Step 1: Identify the first nonzero element in the first row and, if necessary, scale the row to create a leading 1.
Step 2: Use the leading 1 to eliminate the entries below it by adding/subtracting a multiple of the first row from the rows beneath.
Step 3: Move to the next row and repeat the process ensuring each leading 1 is to the right of the one in the previous row.
Final Answer: The matrix is now in row-echelon form with zeros below each leading 1.

Applying Elementary Row Operations

QUESTION

Once the augmented matrix is in row-echelon form, how do you find the solution to the system?

STEP-BY-STEP ANSWER:

Step 1: Start with the last nonzero row; solve for the corresponding variable.
Step 2: Substitute that value into the preceding rows to eliminate the solved variable.
Step 3: Continue back-substitution until all variables are solved.
Final Answer: The system’s solution is found by back-substitution.

Solving via Gaussian Elimination

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Common Mistakes

  • Failing to correctly count the number of rows and columns, thus misidentifying the order of the matrix.
  • Neglecting to include zeros for missing coefficients when forming an augmented matrix.
  • Making arithmetic errors during elementary row operations and not carefully tracking each step.
  • Confusing row-echelon form with reduced row-echelon form; not ensuring that every column with a leading 1 has zeros both above and below it.
  • Overlooking the importance of the order of operations in Gaussian and Gauss-Jordan elimination methods.