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Find the product AB, where A and B are the following matrices: 1 0 a 1 0. A 0 0 1 B = -1 0. 1 1 1

Find the product AB, where A and B are the following matrices: 1 0 a 1 0. A 0 0 1 B = -1 0. 1 1 1

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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To find the product \( AB \), where \( A \) and \( B \) are given matrices, follow these steps: Matrix \( A \) is: \[ A = \begin{bmatrix} 1 & 0 & a \\ 0 & 0 & 1 \\ 0 & 1 & b \end{bmatrix} \] Matrix \( B \) is: \[ B = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ 1 & 1 \end{bmatrix} \] The product \( AB \) is calculated by performing matrix multiplication, where each element of the resulting matrix is the dot product of rows from matrix \( A \) with columns from matrix \( B \). **To calculate each element of the resulting matrix:** - Element (1,1): \((1 \times 1) + (0 \times 0) + (a \times 1)\) - Element (1,2): \((1 \times 0) + (0 \times -1) + (a \times 1)\) - Element (2,1): \((0 \times 1) + (0 \times 0) + (1 \times 1)\) - Element (2,2): \((0 \times 0) + (0 \times -1) + (1 \times 1)\) - Element (3,1): \((0 \times 1) + (1 \times 0) + (b \times 1)\) - Element (3,2): \((0 \times 0) + (1 \times -1) + (b \times 1)\) After performing the calculations, the product matrix \( AB \) is: \[ AB = \begin{bmatrix} 1 + a & a \\ 1 & 1 \\ b & b - 1 \end{bmatrix} \] This matrix represents the result of multiplying matrices \( A \) and \( B \).
Transcribed Image Text:To find the product \( AB \), where \( A \) and \( B \) are given matrices, follow these steps: Matrix \( A \) is: \[ A = \begin{bmatrix} 1 & 0 & a \\ 0 & 0 & 1 \\ 0 & 1 & b \end{bmatrix} \] Matrix \( B \) is: \[ B = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ 1 & 1 \end{bmatrix} \] The product \( AB \) is calculated by performing matrix multiplication, where each element of the resulting matrix is the dot product of rows from matrix \( A \) with columns from matrix \( B \). **To calculate each element of the resulting matrix:** - Element (1,1): \((1 \times 1) + (0 \times 0) + (a \times 1)\) - Element (1,2): \((1 \times 0) + (0 \times -1) + (a \times 1)\) - Element (2,1): \((0 \times 1) + (0 \times 0) + (1 \times 1)\) - Element (2,2): \((0 \times 0) + (0 \times -1) + (1 \times 1)\) - Element (3,1): \((0 \times 1) + (1 \times 0) + (b \times 1)\) - Element (3,2): \((0 \times 0) + (1 \times -1) + (b \times 1)\) After performing the calculations, the product matrix \( AB \) is: \[ AB = \begin{bmatrix} 1 + a & a \\ 1 & 1 \\ b & b - 1 \end{bmatrix} \] This matrix represents the result of multiplying matrices \( A \) and \( B \).
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