https://file.aiquickdraw.com/mathbot/file/1736822261485hsfde4g9.jpg

The image appears to contain a matrix with the question about its rank. I will analyze the content and help you calculate the rank of the matrix.

The matrix given in the image is:
\[
\begin{pmatrix}
1 & 2 & 3 & 0 \\
0 & 1 & 1 & 1 \\
1 & 3 & 4 & 1
\end{pmatrix}
\]

To determine the rank of this matrix, we will use row reduction (Gaussian elimination) to transform it into row echelon form or reduced row echelon form.

I will proceed with the calculation now. numpy as np

# Define the matrix from the image
matrix = np.array([[1, 2, 3, 0],
                   [0, 1, 1, 1],
                   [1, 3, 4, 1]])

# Perform Gaussian elimination (row echelon form)
def row_echelon_form(matrix):
    m, n = matrix.shape
    rank = 0
    for i in range(m):
        if matrix[i, rank] != 0:
            # Normalize the pivot row
            matrix[i] = matrix[i] / matrix[i, rank]
            for j in range(i+1, m):
                matrix[j] = matrix[j] - matrix[j, rank] * matrix[i]
            rank += 1
        else:
            # Look for a non-zero element in the current column
            for k in range(i+1, m):
                if matrix[k, rank] != 0:
                    matrix[[i, k]] = matrix[[k, i]]  # Swap rows
                    matrix[i] = matrix[i] / matrix[i, rank]
                    for j in range(i+1, m):
                        matrix[j] = matrix[j] - matrix[j, rank] * matrix[i]
                    rank += 1
                    break
    return matrix, rank

# Get the row echelon form and rank of the matrix
ref_matrix, matrix_rank = row_echelon_form(matrix.copy())
ref_matrix, matrix_rankAfter performing the row reduction, we obtain the row echelon form of the matrix as:

\[
\begin{pmatrix}
1 & 2 & 3 & 0 \\
0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

The rank of this matrix is **2**, as there are two non-zero rows in the row echelon form.

Would you like further details or explanations on this? 

Here are some related questions for you:
1. How do you calculate the rank of a matrix from its row echelon form?
2. What is the difference between row echelon form and reduced row echelon form?
3. Can the rank of a matrix change when performing different types of row operations?
4. What are the practical applications of calculating the rank of a matrix?
5. How do you find the rank of a matrix using the determinant?

Tip: When working with matrix rank, always ensure to simplify the matrix using row operations until you have the maximum number of linearly independent rows.

Find the rank of the matrix using Gaussian elimination: 
\(
\begin{pmatrix}
1 & 2 & 3 & 0 \\
0 & 1 & 1 & 1 \\
1 & 3 & 4 & 1
\end{pmatrix}
\)

Learn how to calculate the rank of a 3x4 matrix using Gaussian elimination. Step-by-step solution to reduce the matrix to row echelon form and find the rank. Understand core concepts in linear algebra and their practical applications.

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