https://file.aiquickdraw.com/mathbot/file/1734341361890vq4vao9s.pdf

The first question in the document asks to **reduce the given matrices to their echelon form and find their rank**. Let's solve part **(a)** of the question. The matrix provided is:

\[
A = \begin{bmatrix}
2 & 3 & -1 & -1 \\
1 & -1 & -2 & -4 \\
3 & 1 & 3 & -2 \\
6 & 3 & 0 & -7
\end{bmatrix}
\]

### Steps to Solve:
1. **Row reduce the matrix to echelon form**:
   - Start by making the first pivot element (top-left) equal to 1 by dividing the first row by 2.
   - Use the first row to eliminate the entries below the pivot in the first column.
   - Repeat this process for subsequent pivot elements in columns 2, 3, etc.

2. **Determine the rank**:
   - The rank is the number of non-zero rows in the echelon form.

Let me calculate the echelon form and the rank. sympy as sp

# Define the matrix A
A = sp.Matrix([
    [2, 3, -1, -1],
    [1, -1, -2, -4],
    [3, 1, 3, -2],
    [6, 3, 0, -7]
])

# Compute the echelon form
echelon_form = A.rref()  # rref() returns (reduced echelon form, pivot columns)
echelon_matrix = echelon_form[0]
rank = len(echelon_form[1])  # The rank is the number of pivot columns

echelon_matrix, rankThe echelon form of the given matrix is:

\[
\begin{bmatrix}
1 & 0 & 0 & -\frac{5}{3} \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & \frac{2}{3} \\
0 & 0 & 0 & 0
\end{bmatrix}
\]

### Rank of the Matrix:
The rank of the matrix is **3**, as there are 3 non-zero rows in the echelon form.

---

Let me know if you'd like detailed steps for the row-reduction or need help with the other parts of the question. Here are some related questions you might find useful:

1. How do you interpret matrix rank in terms of linear independence?
2. What is the difference between row echelon form and reduced row echelon form?
3. How do you determine if a system of equations is consistent using the rank of a matrix?
4. Can a non-square matrix have a full rank? If yes, how is it determined?
5. What is the significance of pivot columns in row reduction?

### Tip:
Always check your row operations carefully to avoid arithmetic mistakes, especially when working with fractions.

Understand how to reduce matrices to echelon form and calculate their rank. This comprehensive guide explains the row reduction process for matrices with examples, including calculating the rank from the echelon form. Ideal for undergraduate students studying linear algebra.

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