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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![### Reducing a Matrix Using Row Operations
The goal of this lesson is to show how to use row operations to transform a given matrix into its reduced form.
#### Initial Matrix
We begin with the following matrix, which we'll denote as matrix A:
\[ A = \begin{bmatrix}
1 & 2 & -2 & | & -1 \\
0 & 5 & -15 & | & 1 \\
0 & -1 & 3 & | & -\frac{1}{5}
\end{bmatrix} \]
Here, each row is separated by horizontal lines, and the matrix includes an augmented column (represented by the vertical bar) for solving a system of linear equations.
#### Process of Row Reduction
We aim to use row operations to simplify matrix \( A \) into its reduced row echelon form (RREF). Row operations include:
- Swapping rows.
- Multiplying a row by a non-zero scalar.
- Adding or subtracting the multiples of one row to or from another row.
#### Resultant Matrix
After applying the appropriate row operations (which will be detailed below), we transform matrix \( A \) into its reduced form.
\[ \begin{bmatrix}
1 & 2 & -2 & | & -1 \\
0 & 5 & -15 & | & 1 \\
0 & -1 & 3 & | & -\frac{1}{5}
\end{bmatrix}
\xrightarrow{\text{Row Operations}}
\begin{bmatrix}
\boxed{} & \boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} & \boxed{}
\end{bmatrix}
\]
In the image, the boxes in the resultant matrix represent the final values which we get after performing the row operations. Since the values aren't filled in the image, they indicate placeholders for the calculated values.
#### Explanation of the Diagram
- The diagram shows the matrix \( A \) and the symbol \( \sim \), indicating that matrix \( A \) will be converted or is equivalent to its reduced form through row operations.
- The boxes in the reduced matrix signify the positions where numerical values will be placed after completing the row reduction process.
#### Conclusion
To complete this transformation, specific row operations need to be performed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d9df1eb-9cbe-4888-87d6-c14249115dd3%2F604a8bed-6991-4552-b486-7d9d1e36adb3%2Fzg7jodr_processed.png&w=3840&q=75)
Transcribed Image Text:### Reducing a Matrix Using Row Operations
The goal of this lesson is to show how to use row operations to transform a given matrix into its reduced form.
#### Initial Matrix
We begin with the following matrix, which we'll denote as matrix A:
\[ A = \begin{bmatrix}
1 & 2 & -2 & | & -1 \\
0 & 5 & -15 & | & 1 \\
0 & -1 & 3 & | & -\frac{1}{5}
\end{bmatrix} \]
Here, each row is separated by horizontal lines, and the matrix includes an augmented column (represented by the vertical bar) for solving a system of linear equations.
#### Process of Row Reduction
We aim to use row operations to simplify matrix \( A \) into its reduced row echelon form (RREF). Row operations include:
- Swapping rows.
- Multiplying a row by a non-zero scalar.
- Adding or subtracting the multiples of one row to or from another row.
#### Resultant Matrix
After applying the appropriate row operations (which will be detailed below), we transform matrix \( A \) into its reduced form.
\[ \begin{bmatrix}
1 & 2 & -2 & | & -1 \\
0 & 5 & -15 & | & 1 \\
0 & -1 & 3 & | & -\frac{1}{5}
\end{bmatrix}
\xrightarrow{\text{Row Operations}}
\begin{bmatrix}
\boxed{} & \boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} & \boxed{}
\end{bmatrix}
\]
In the image, the boxes in the resultant matrix represent the final values which we get after performing the row operations. Since the values aren't filled in the image, they indicate placeholders for the calculated values.
#### Explanation of the Diagram
- The diagram shows the matrix \( A \) and the symbol \( \sim \), indicating that matrix \( A \) will be converted or is equivalent to its reduced form through row operations.
- The boxes in the reduced matrix signify the positions where numerical values will be placed after completing the row reduction process.
#### Conclusion
To complete this transformation, specific row operations need to be performed.
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