Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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**Use the inverse matrices to find \( (AB)^{-1} \), \( (A^T)^{-1} \), and \( (2A)^{-1} \).** Given: \[ A^{-1} = \begin{bmatrix} \frac{1}{2} & 1 & \frac{3}{4} \\ 1 & -2 & \frac{1}{4} \\ -2 & 1 & \frac{1}{2} \end{bmatrix}, \quad B^{-1} = \begin{bmatrix} 2 & 1 & 4 \\ 2 & \frac{1}{2} & 2 \\ 1 & -\frac{3}{4} & 2 \end{bmatrix} \] **(a) \( (AB)^{-1} \)** Diagram with three rows and three columns of empty boxes indicating the positions for each element of the resulting inverse matrix. There are green arrows pointing right and downward, implying the order and direction for filling in the matrix. **(b) \( (A^T)^{-1} \)** Another similar diagram with empty boxes structured in a 3x3 matrix format, with green arrows indicating the direction for determining the elements of the inverse of the transpose of matrix \( A \). **(c) \( (2A)^{-1} \)** The third diagram, following the same format. It shows empty boxes for a 3x3 matrix, with green arrows pointing right and downward to help guide solving for the inverse of \( 2A \). In each sub-section, students are expected to compute the inverse matrices based on the provided inverse matrices of \( A \) and \( B \), applying the necessary mathematical rules and operations for matrix inverses.
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Transcribed Image Text:**Use the inverse matrices to find \( (AB)^{-1} \), \( (A^T)^{-1} \), and \( (2A)^{-1} \).** Given: \[ A^{-1} = \begin{bmatrix} \frac{1}{2} & 1 & \frac{3}{4} \\ 1 & -2 & \frac{1}{4} \\ -2 & 1 & \frac{1}{2} \end{bmatrix}, \quad B^{-1} = \begin{bmatrix} 2 & 1 & 4 \\ 2 & \frac{1}{2} & 2 \\ 1 & -\frac{3}{4} & 2 \end{bmatrix} \] **(a) \( (AB)^{-1} \)** Diagram with three rows and three columns of empty boxes indicating the positions for each element of the resulting inverse matrix. There are green arrows pointing right and downward, implying the order and direction for filling in the matrix. **(b) \( (A^T)^{-1} \)** Another similar diagram with empty boxes structured in a 3x3 matrix format, with green arrows indicating the direction for determining the elements of the inverse of the transpose of matrix \( A \). **(c) \( (2A)^{-1} \)** The third diagram, following the same format. It shows empty boxes for a 3x3 matrix, with green arrows pointing right and downward to help guide solving for the inverse of \( 2A \). In each sub-section, students are expected to compute the inverse matrices based on the provided inverse matrices of \( A \) and \( B \), applying the necessary mathematical rules and operations for matrix inverses.
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