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
19
![**Matrix Row Operations to Reduced Form**
---
### Given Matrix:
Use row operations to change the matrix to reduced form:
\[
\begin{bmatrix}
1 & 0 & -5 & \vert & 1 \\
0 & 1 & 6 & \vert & 0 \\
0 & 0 & 4 & \vert & -8 \\
\end{bmatrix}
\]
---
### Intermediate Step:
Applying row operations to transform the matrix:
\[
\begin{bmatrix}
1 & 0 & -5 & \vert & 1 \\
0 & 1 & 6 & \vert & 0 \\
0 & 0 & 4 & \vert & -8 \\
\end{bmatrix}
\approx
\begin{bmatrix}
\quad \, \boxdot & \, \boxdot & \, \boxdot & \,\vert & \, \boxdot \\
\quad \, \boxdot & \, \boxdot & \, \boxdot & \,\vert & \, \boxdot \\
\quad \, \boxdot & \, \boxdot & \, \boxdot & \,\vert & \, \boxdot \\
\end{bmatrix}
\]
In the second matrix, the operations to achieve the reduced form are symbolized by an approximate (∼) sign. Each \(\boxdot\) represents the placeholder for new matrix elements that will be obtained after performing the necessary row operations.
**Key Points:**
- The first matrix (above) represents the augmented form \( [A \vert b] \) of a system of linear equations.
- The goal is to simplify this matrix using row operations to achieve the reduced form (or RREF).
- Achieving RREF involves making all the leading coefficients (also called pivots) ones, and all other elements in their respective columns zeros.
Row operations typically include:
1. Swapping two rows.
2. Multiplying a row by a non-zero scalar.
3. Adding or subtracting a multiple of one row to another row.
Understanding matrix transformations is crucial for solving linear equations efficiently.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d9df1eb-9cbe-4888-87d6-c14249115dd3%2F1130f2d6-6357-44b5-a3d0-1f46a80792d7%2F8i7mcuj_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Row Operations to Reduced Form**
---
### Given Matrix:
Use row operations to change the matrix to reduced form:
\[
\begin{bmatrix}
1 & 0 & -5 & \vert & 1 \\
0 & 1 & 6 & \vert & 0 \\
0 & 0 & 4 & \vert & -8 \\
\end{bmatrix}
\]
---
### Intermediate Step:
Applying row operations to transform the matrix:
\[
\begin{bmatrix}
1 & 0 & -5 & \vert & 1 \\
0 & 1 & 6 & \vert & 0 \\
0 & 0 & 4 & \vert & -8 \\
\end{bmatrix}
\approx
\begin{bmatrix}
\quad \, \boxdot & \, \boxdot & \, \boxdot & \,\vert & \, \boxdot \\
\quad \, \boxdot & \, \boxdot & \, \boxdot & \,\vert & \, \boxdot \\
\quad \, \boxdot & \, \boxdot & \, \boxdot & \,\vert & \, \boxdot \\
\end{bmatrix}
\]
In the second matrix, the operations to achieve the reduced form are symbolized by an approximate (∼) sign. Each \(\boxdot\) represents the placeholder for new matrix elements that will be obtained after performing the necessary row operations.
**Key Points:**
- The first matrix (above) represents the augmented form \( [A \vert b] \) of a system of linear equations.
- The goal is to simplify this matrix using row operations to achieve the reduced form (or RREF).
- Achieving RREF involves making all the leading coefficients (also called pivots) ones, and all other elements in their respective columns zeros.
Row operations typically include:
1. Swapping two rows.
2. Multiplying a row by a non-zero scalar.
3. Adding or subtracting a multiple of one row to another row.
Understanding matrix transformations is crucial for solving linear equations efficiently.
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