Math Problem Statement
Find the rank of the matrix using Gaussian elimination:
\(
\begin{pmatrix}
1 & 2 & 3 & 0 \\
0 & 1 & 1 & 1 \\
1 & 3 & 4 & 1
\end{pmatrix}
\)
Solution
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:
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Gaussian Elimination
Row Echelon Form
Formulas
Row reduction operations for Gaussian elimination
Rank of a matrix = Number of non-zero rows in row echelon form
Theorems
Rank-Nullity Theorem
Suitable Grade Level
Undergraduate - Year 1