### Determining the Rank of a MatrixGiven 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 Matrix1. **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.### ResultAfter 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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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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Question

### 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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