Math Problem Statement
Exercise 5. Determine the rank of A = (aij )4×5 given by A = 3 2 0 5 0 3 −2 3 6 −1 2 0 1 5 −3 1 6 −4 −1 4
Solution
To determine the rank of the given matrix , we need to find the maximum number of linearly independent rows or columns of the matrix. Here's the step-by-step process:
Step 1: Write the matrix
The matrix is:
3 & 2 & 0 & 5 & 0 \\ 3 & -2 & 3 & 6 & -1 \\ 2 & 0 & 1 & 5 & -3 \\ 1 & 6 & -4 & -1 & 4 \end{bmatrix}$$ ### Step 2: Apply row reduction We use Gaussian elimination to reduce the matrix to row echelon form. This involves making the leading coefficients below each pivot element zero by row operations. 1. **Make the first pivot** (first element of the first row) $$3$$, then eliminate the entries below it. - Subtract $$\frac{3}{3} \times \text{Row 1}$$ from Row 2. - Subtract $$\frac{2}{3} \times \text{Row 1}$$ from Row 3. - Subtract $$\frac{1}{3} \times \text{Row 1}$$ from Row 4. 2. Perform similar operations for subsequent rows to ensure a triangular form. Eliminate entries below each pivot. ### Step 3: Identify non-zero rows After performing row reduction, the rank of the matrix is equal to the number of non-zero rows in its row echelon form. ### Calculation Let me compute this explicitly. numpy as np # Define the matrix A A = np.array([ [3, 2, 0, 5, 0], [3, -2, 3, 6, -1], [2, 0, 1, 5, -3], [1, 6, -4, -1, 4] ]) # Perform row reduction (find rank) rank_A = np.linalg.matrix_rank(A) rank_AThe rank of the given matrix $$A$$ is $$\mathbf{3}$$. This means there are three linearly independent rows (or columns) in $$A$$. Would you like a detailed explanation of the row reduction steps or more context on the concept of rank? Here are some related questions for further exploration: 1. How does the rank relate to the number of pivot elements in a matrix? 2. Can a matrix have a rank greater than its smallest dimension (rows or columns)? 3. What does a rank of 3 imply about the null space of $$A$$? 4. How would the rank of $$A$$ change if an extra row or column is added? 5. How is rank used in solving systems of linear equations? **Tip:** To verify your row reduction steps, use a determinant-based approach on smaller submatrices to confirm independence.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Reduction
Gaussian Elimination
Formulas
Matrix rank = number of non-zero rows in row echelon form
Row operations: swapping, scaling, and row addition
Theorems
Rank Theorem
Echelon Form
Suitable Grade Level
Undergraduate