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 \( B \).

Answered: Find the product AB, where A and B are…

Math

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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

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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Question

100%

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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