Determine the rank of the matrix. 1 -2 2 -3] 6) 2 -4 7 -2 -3 6 -6 9
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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![### Determining the Rank of a Matrix
Given the following matrix, determine its rank:
\[
\begin{pmatrix}
1 & -2 & 2 & -3 \\
2 & -4 & 7 & -2 \\
-3 & 6 & -6 & 9
\end{pmatrix}
\]
### Steps to Determine the Rank of the Matrix
1. **Construct the Augmented Matrix**:
- Write down the matrix in its given form.
2. **Apply Row Operations**:
- Use elementary row operations to transform the matrix into its row echelon form (REF) or reduced row echelon form (RREF). Row operations include row swapping, scaling, and row addition/subtraction.
3. **Identify the Pivot Columns**:
- Once in echelon form, count the number of non-zero rows (pivot rows). Each pivot row corresponds to a leading 1 in a different column.
4. **Determine the Rank**:
- The rank of the matrix is the number of pivot rows in the REF or RREF.
### Example Solution
- Start with the given matrix and perform row operations to simplify it:
\[
\begin{pmatrix}
1 & -2 & 2 & -3 \\
2 & -4 & 7 & -2 \\
-3 & 6 & -6 & 9
\end{pmatrix}
\]
- Sample row operation step-by-step:
- Replace the second row with (Row 2 - 2*Row 1)
- Replace the third row with (Row 3 + 3*Row 1)
- Further simplify the resulting matrix.
- Continue performing row operations until you reach the REF or RREF.
### Result
After reaching the echelon form, count the number of leading 1's (pivot positions):
\[
\text{Rank} = \text{Number of pivot positions}
\]
This matrix will generally have specific row operations, which might look something like this in the row echelon form:
\[
\begin{pmatrix}
1 & -2 & 2 & -3 \\
0 & 0 & 3 & -4 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]
From this form, we can see that there are 2 non-zero rows, thus indicating that the rank of the matrix is](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e51c4f9-7707-4851-92b0-3f7eb5342ba2%2F836dd055-8b3f-4688-89a5-e4866bfdf50e%2Fq3maf2_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining the Rank of a Matrix
Given the following matrix, determine its rank:
\[
\begin{pmatrix}
1 & -2 & 2 & -3 \\
2 & -4 & 7 & -2 \\
-3 & 6 & -6 & 9
\end{pmatrix}
\]
### Steps to Determine the Rank of the Matrix
1. **Construct the Augmented Matrix**:
- Write down the matrix in its given form.
2. **Apply Row Operations**:
- Use elementary row operations to transform the matrix into its row echelon form (REF) or reduced row echelon form (RREF). Row operations include row swapping, scaling, and row addition/subtraction.
3. **Identify the Pivot Columns**:
- Once in echelon form, count the number of non-zero rows (pivot rows). Each pivot row corresponds to a leading 1 in a different column.
4. **Determine the Rank**:
- The rank of the matrix is the number of pivot rows in the REF or RREF.
### Example Solution
- Start with the given matrix and perform row operations to simplify it:
\[
\begin{pmatrix}
1 & -2 & 2 & -3 \\
2 & -4 & 7 & -2 \\
-3 & 6 & -6 & 9
\end{pmatrix}
\]
- Sample row operation step-by-step:
- Replace the second row with (Row 2 - 2*Row 1)
- Replace the third row with (Row 3 + 3*Row 1)
- Further simplify the resulting matrix.
- Continue performing row operations until you reach the REF or RREF.
### Result
After reaching the echelon form, count the number of leading 1's (pivot positions):
\[
\text{Rank} = \text{Number of pivot positions}
\]
This matrix will generally have specific row operations, which might look something like this in the row echelon form:
\[
\begin{pmatrix}
1 & -2 & 2 & -3 \\
0 & 0 & 3 & -4 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]
From this form, we can see that there are 2 non-zero rows, thus indicating that the rank of the matrix is
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