To determine the rank of the matrix $B$ displayed in the image, we need to find the number of linearly independent rows or columns. The matrix $B$ is given as:

$B = \begin{pmatrix} 6 & 1 & 3 \\ 6 & 6 & 5 \\ 2 & 4 & 7 \\ 2 & 1 & 0 \\ 7 & 4 & 7 \end{pmatrix}$

The steps to determine the rank involve row reduction to find the echelon form of the matrix, then counting the number of non-zero rows.

Step-by-Step Solution

1. Set up matrix for row reduction.
2. Perform elementary row operations to create zeros below the leading 1s in each column.
3. Identify the non-zero rows in the echelon form.

Let's proceed with the row reduction for matrix $B$.

After simplifying, I’ll calculate the rank.import numpy as np

Define matrix B as given in the image

B = np.array([ [6, 1, 3], [6, 6, 5], [2, 4, 7], [2, 1, 0], [7, 4, 7] ])

Perform row echelon reduction using numpy's rank function to determine the rank

rank_B = np.linalg.matrix_rank(B) rank_B33successThe rank of the matrix $B$ is 3. This means that there are 3 linearly independent rows (or columns) in this matrix.

Would you like further details on the row reduction process or have any other questions?

Here are some related questions to consider:

1. How do we determine if rows or columns are linearly independent?
2. What is the significance of the rank of a matrix in linear algebra?
3. Can we find the rank of a matrix without performing row reduction?
4. How does the rank of a matrix relate to the solutions of linear systems?
5. What does it mean if a matrix has full rank?

Tip: The rank of a matrix is always less than or equal to the smallest dimension of the matrix (number of rows or columns).