https://file.aiquickdraw.com/mathbot/file/1724506095640muuhq6tw.jpg

To determine the rank of the matrix \( A \), I will follow a step-by-step procedure:

1. **Matrix Representation**: First, I will extract the matrix from the image. Here's the matrix \( A \):

\[
A = \begin{pmatrix}
6 & 1 & 3 & 8 \\
4 & 2 & 6 & -1 \\
10 & 3 & 9 & 7 \\
16 & 4 & 12 & 15
\end{pmatrix}
\]

2. **Row Reduction**: To find the rank, I will perform row reduction (Gaussian elimination) on the matrix to convert it into row echelon form (REF). The rank is the number of non-zero rows in this echelon form.

I will now perform these calculations and determine the rank of the matrix \( A \).import numpy as np
from scipy.linalg import lu

# Defining the matrix A
A = np.array([[6, 1, 3, 8],
              [4, 2, 6, -1],
              [10, 3, 9, 7],
              [16, 4, 12, 15]])

# Performing row reduction and determining the rank
rank_A = np.linalg.matrix_rank(A)

rank_AThe rank of the matrix \( A \) is \( 2 \). 

This means there are 2 linearly independent rows or columns in the matrix.

Would you like further details or have any questions?

Here are some related questions to explore:
1. How is the rank of a matrix related to the system of linear equations?
2. What is the difference between row echelon form and reduced row echelon form?
3. How can we interpret rank in terms of the dimensionality of the solution space?
4. What does it mean if a matrix has full rank?
5. How does row reduction help in solving linear systems?

**Tip**: The rank of a matrix gives you insight into the linear dependence of its rows or columns and plays a key role in determining the solution space of a system of equations.

Learn how to determine the rank of matrix A using Gaussian elimination and matrix rank calculation. Understand the significance of matrix rank in linear algebra. Suitable for undergraduate students and anyone interested in matrix theory.

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