Textbook Question
Write each matrix equation as a system of linear equations without matrices.
Ch. 6 - Matrices and Determinants
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 7, Problem 35
In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant.
Verified step by step guidance1
Step 1: Write down the given 3x3 determinant:
\[\left| \begin{array}{ccc} 0.5 & 7 & 5 \\ 0.5 & 3 & 9 \\ 0.5 & 1 & 3 \end{array} \right|\]
Step 2: Use the alternative method for evaluating third-order determinants, which involves expanding along the first row or using the rule of Sarrus. Here, we will use the rule of Sarrus:
- Repeat the first two columns to the right of the matrix:
\[\begin{array}{ccc|cc} 0.5 & 7 & 5 & 0.5 & 7 \\ 0.5 & 3 & 9 & 0.5 & 3 \\ 0.5 & 1 & 3 & 0.5 & 1 \end{array}\]
Step 3: Calculate the sum of the products of the diagonals from top-left to bottom-right:
\[ (0.5 \times 3 \times 3) + (7 \times 9 \times 0.5) + (5 \times 0.5 \times 1) \]
Step 4: Calculate the sum of the products of the diagonals from bottom-left to top-right:
\[ (0.5 \times 3 \times 5) + (1 \times 9 \times 0.5) + (3 \times 0.5 \times 7) \]
Step 5: Subtract the sum from Step 4 from the sum in Step 3 to find the value of the determinant:
\[ \text{Determinant} = \left[ (0.5 \times 3 \times 3) + (7 \times 9 \times 0.5) + (5 \times 0.5 \times 1) \right] - \left[ (0.5 \times 3 \times 5) + (1 \times 9 \times 0.5) + (3 \times 0.5 \times 7) \right] \]
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Third-Order Determinants
A third-order determinant is a scalar value calculated from a 3x3 matrix. It helps determine properties like matrix invertibility and solutions to systems of equations. The determinant is computed using specific formulas or methods, such as expansion by minors or the alternative method.
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Determinants of 3×3 Matrices
Alternative Method for Evaluating Determinants
The alternative method, often called the Rule of Sarrus, is a shortcut for finding the determinant of a 3x3 matrix. It involves summing the products of diagonals from left to right and subtracting the products of diagonals from right to left, simplifying the calculation process.
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Matrix Representation of Systems of Equations
Matrices can represent systems of linear equations compactly, where each row corresponds to an equation and each column to a variable or constant. Evaluating the determinant of the coefficient matrix helps determine if the system has a unique solution, infinite solutions, or none.
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Related Practice
Textbook Question
Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
Textbook Question
In Exercises 37–44, use Cramer's Rule to solve each system.
Textbook Question
In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 4 2 2 3 4 A = 6 1 B = 3 5 - 1 - 2 0
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Textbook Question
In Exercises 37–38, find the products and to determine whether B is the multiplicative inverse of A.
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Textbook Question
In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 2 - 3 1 - 1 - 1 1 A = B = 1 1 - 2 1 5 4 10 5