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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 5
Capitolo 7, Problema 5

Find the products AB and BA to determine whether B is the multiplicative inverse of A.
A=[−2132−12],B=[1234]A = \(\begin{bmatrix}\) -2 & 1 \\ \(\frac{3}{2}\) & -\(\frac{1}{2}\) \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 1 & 2 \\ 3 & 4 \(\end{bmatrix}\)

Guida verificata passo dopo passo
1
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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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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Introduction to Matrices
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