Math Problem Statement
Find the rank of the given matrix: A = [[7, -7, 5, 1, 5], [2, 8, 3, -4, 6], [1, -8, 1, 4, 2]].
Solution
The task here is to find the rank of the given matrix. Let's denote the matrix as:
7 & -7 & 5 & 1 & 5 \\ 2 & 8 & 3 & -4 & 6 \\ 1 & -8 & 1 & 4 & 2 \end{pmatrix}$$ To compute the rank of the matrix, we reduce it to its row echelon form or reduced row echelon form and count the number of non-zero rows. I will calculate the rank of this matrix step by step. numpy as np # Define the matrix matrix = np.array([ [7, -7, 5, 1, 5], [2, 8, 3, -4, 6], [1, -8, 1, 4, 2] ]) # Compute the rank of the matrix rank = np.linalg.matrix_rank(matrix) rankThe rank of the given matrix is **3**. This means there are 3 linearly independent rows in the matrix, which is the maximum rank possible for a $$3 \times 5$$ matrix. Would you like me to explain the steps for finding the rank in detail, such as row reducing the matrix manually, or do you have any related questions? ### Related questions: 1. What is the definition of matrix rank, and how is it applied in linear algebra? 2. What are the methods to determine the rank of a matrix manually? 3. Can the rank be used to determine if a system of equations is consistent or inconsistent? 4. How does the rank relate to the dimensions of the column and null spaces of a matrix? 5. What happens if a square matrix has a rank less than its size? ### Tip: Always check the rank of a matrix to understand the dimensions of its row and column spaces, especially when solving systems of linear equations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Echelon Form
Linear Independence
Formulas
Rank = Number of linearly independent rows or columns
Theorems
Rank-Nullity Theorem
Suitable Grade Level
Grades 11-12 and Undergraduate