Given the matrix 222-6-2 1 2 6 -1 -6 Use elementary row operations to carry it to a matrix that is a) In reduced row-echelon form: 0000 Reduced row-echelon form: 0 0 0 0 0000
Given the matrix 222-6-2 1 2 6 -1 -6 Use elementary row operations to carry it to a matrix that is a) In reduced row-echelon form: 0000 Reduced row-echelon form: 0 0 0 0 0000
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Gaussian Elimination and Row Reduction to Row-Echelon Form
#### Given Matrix
Consider the following matrix:
\[
\begin{bmatrix}
2 & 2 & -6 & -2 \\
1 & 2 & 6 & -1 \\
2 & -1 & -6 & -6
\end{bmatrix}
\]
We aim to use elementary row operations to convert this given matrix into its **Reduced Row-Echelon Form (RREF)**.
#### Reduced Row-Echelon Form
The RREF of a matrix is a matrix form that satisfies the following conditions:
1. The leading entry in each nonzero row is 1 (known as a pivot or leading one).
2. Each leading one is the only nonzero entry in its column.
3. The leading one in a row is to the right of any leading ones in the previous rows.
4. Rows with all zero elements, if any, are below rows which contain a leading one.
Upon performing row operations on the given matrix, its Reduced Row-Echelon Form is achieved as:
\[
\begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]
### Explanation of Graph or Diagram
In this case, there is no graphical representation or diagram. The main focus is on the transformation of the matrix into its reduced form using elementary row operations.
### Interpretation
The resulting matrix of all zeros suggests that the given system of equations (represented by the original matrix) does not have unique solutions or has many solutions, typically indicating that the equations are dependent. This generally implies that there are free variables in the system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd936c6c0-8e3b-4a92-9646-2c460fec57ae%2F550d328a-f5d2-4873-badf-7c9323ddaef8%2Fc9tm0zq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Gaussian Elimination and Row Reduction to Row-Echelon Form
#### Given Matrix
Consider the following matrix:
\[
\begin{bmatrix}
2 & 2 & -6 & -2 \\
1 & 2 & 6 & -1 \\
2 & -1 & -6 & -6
\end{bmatrix}
\]
We aim to use elementary row operations to convert this given matrix into its **Reduced Row-Echelon Form (RREF)**.
#### Reduced Row-Echelon Form
The RREF of a matrix is a matrix form that satisfies the following conditions:
1. The leading entry in each nonzero row is 1 (known as a pivot or leading one).
2. Each leading one is the only nonzero entry in its column.
3. The leading one in a row is to the right of any leading ones in the previous rows.
4. Rows with all zero elements, if any, are below rows which contain a leading one.
Upon performing row operations on the given matrix, its Reduced Row-Echelon Form is achieved as:
\[
\begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]
### Explanation of Graph or Diagram
In this case, there is no graphical representation or diagram. The main focus is on the transformation of the matrix into its reduced form using elementary row operations.
### Interpretation
The resulting matrix of all zeros suggests that the given system of equations (represented by the original matrix) does not have unique solutions or has many solutions, typically indicating that the equations are dependent. This generally implies that there are free variables in the system.
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