Domanda del libro di testo
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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TOC.tableOfContents
- 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
Capitolo 7, Problema 7
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
Guida verificata passo dopo passo1
Step 1: Identify the matrices A and B. Matrix A is given as \(A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}\) and matrix B is \(B = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}\).
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. Formally, \((AB)_{ij} = \sum_{k=1}^3 A_{ik} B_{kj}\).
Step 3: Perform the multiplication for each element of the product matrix \(AB\) by calculating the dot products row by row and column by column.
Step 4: Repeat the process to find the product \(BA\) by multiplying matrix B by matrix A, using the same dot product method for each element: \((BA)_{ij} = \sum_{k=1}^3 B_{ik} A_{kj}\).
Step 5: After finding both products \(AB\) and \(BA\), compare each with the identity matrix \(I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\). If both $AB = I$ and $BA = I$, then matrix B is the multiplicative inverse of matrix A.
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Matrix Multiplication
Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. The order of multiplication matters, so AB and BA can yield different results. Understanding this process is essential to compute the products correctly.
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Finding Zeros & Their Multiplicity
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. Verifying both products confirms if B is truly the inverse of A.
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03:42Finding Zeros & Their Multiplicity
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. Recognizing the identity matrix is key to checking inverses.
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