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 \( B \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F704a64b5-5250-41d0-9c29-5aaf5a50e535%2Fd7adcef6-c8fe-4e5c-9382-e5a78ab5c4d1%2Ff0ghv8_processed.jpeg&w=3840&q=75)
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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