Math Problem Statement
Determine the rank of the given 4x5 matrix A:
A = [ [1, 2, 3, 4, 5], [2, 4, 6, 8, 11], [3, 6, 9, 12, 14], [4, 8, 12, 16, 20] ].
Solution
The problem is to determine the rank of the given matrix . The matrix is:
1 & 2 & 3 & 4 & 5 \\ 2 & 4 & 6 & 8 & 11 \\ 3 & 6 & 9 & 12 & 14 \\ 4 & 8 & 12 & 16 & 20 \end{bmatrix}.$$ ### Steps to Find the Rank: 1. **Form the augmented matrix and simplify via row reduction**: We use Gaussian elimination to transform $$A$$ into row echelon form (upper triangular form) or reduced row echelon form. The rank is the number of non-zero rows in this form. 2. **Row reduction**: Observe that: - Row 2 is a multiple of Row 1 (specifically, $$R_2 = 2 \cdot R_1$$). - Row 3 is another multiple of Row 1 ($$R_3 = 3 \cdot R_1$$). - Row 4 is also a multiple of Row 1 ($$R_4 = 4 \cdot R_1$$). Therefore, the entire matrix can be reduced to one non-zero row. ### Conclusion: The rank of matrix $$A$$ is $$r(A) = 1$$. Thus, the correct answer is: **A. $$r(A) = 1$$**. Would you like further details or explanations? Let me know! Here are 5 related questions to deepen your understanding: 1. What is the definition of the rank of a matrix? 2. How does the rank relate to the solution of a system of linear equations? 3. Can the rank of a $$4 \times 5$$ matrix exceed 4? Why or why not? 4. What is the geometric interpretation of matrix rank in terms of vectors? 5. How can rank be computed using the determinant for square matrices? **Tip**: The rank of a matrix is also the maximum number of linearly independent rows or columns.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Reduction
Formulas
Rank of a matrix = number of non-zero rows after row reduction
Theorems
Gaussian Elimination
Linear Independence
Suitable Grade Level
Undergraduate Mathematics or Advanced High School