Textbook Question
Write the augmented matrix for each system of linear equations.
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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 5
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
Verified step by step guidance1
Step 1: Write down the matrices A and B clearly. Matrix A is given by \(A = \begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}\) and matrix B is \(B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\).
Step 2: To find the product \(AB\), multiply matrix A by matrix B. Recall that the element in the \(i^{th}\) row and \(j^{th}\) column of the product matrix is found by taking the dot product of the \(i^{th}\) row of A with the \(j^{th}\) column of B. For example, the element in the first row and first column of \(AB\) is calculated as \((-2)(1) + (1)(3)\).
Step 3: Similarly, calculate all elements of the product matrix \(AB\) by performing the dot products for each position: first row second column, second row first column, and second row second column.
Step 4: Next, find the product \(BA\) by multiplying matrix B by matrix A. Use the same method of dot products, but this time with rows of B and columns of A.
Step 5: After finding both products \(AB\) and \(BA\), compare each product to the identity matrix \(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\). If both \(AB = I\) and \(BA = I\), then matrix B is the multiplicative inverse of matrix A.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Matrix Multiplication
Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. It is essential to correctly compute the products AB and BA to verify if B is the inverse of A. The operation is not commutative, so AB and BA may differ.
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Multiplicative Inverse of a Matrix
A matrix B is the multiplicative inverse of matrix A if both AB and BA equal the identity matrix. This means multiplying A by B returns the identity matrix, which acts like 1 in matrix algebra. Checking both products confirms if B is truly the inverse.
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Identity Matrix
The identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere. It serves as the multiplicative identity in matrix operations, meaning any matrix multiplied by the identity matrix remains unchanged. Verifying AB = I and BA = I confirms the inverse relationship.
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4:35Introduction to Matrices
Related Practice
Textbook Question
In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists.
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Textbook Question
Evaluate each determinant in Exercises 1–10.
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Textbook Question
In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal.
Textbook Question
Write the augmented matrix for each system of linear equations.
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Textbook Question
Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
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