Math Problem Statement
Reduce the following matrix to normal form and find its rank.
Solution
The problem asks to reduce the given matrix to its normal form (Row Echelon Form or Reduced Row Echelon Form) and find its rank. The given matrix is:
6 & 1 & 3 & 8 \\ 4 & 2 & 6 & -1 \\ 10 & 3 & 9 & 7 \\ 16 & 4 & 12 & 15 \end{bmatrix}$$ The operations listed beside the matrix are: - $$A_4 \rightarrow A_4 - (R_1 + R_3)$$ - $$R_3 \rightarrow R_3 - (R_1 + R_2)$$ Let me perform the row operations and find the rank of the matrix step-by-step.The matrix after performing the row operations is: $$\begin{bmatrix} 6 & 1 & 3 & 8 \\ 4 & 2 & 6 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ Converting this to Row Echelon Form (REF), we have: $$\begin{bmatrix} 6 & 1 & 3 & 8 \\ 0 & 8 & 24 & -38 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ The rank of this matrix is **2**, because there are 2 non-zero rows. Would you like further details or clarifications? Here are 5 related questions: 1. What are the steps to find the rank of a matrix in general? 2. How do elementary row operations affect the rank of a matrix? 3. Can you explain the difference between row echelon form (REF) and reduced row echelon form (RREF)? 4. How would the rank change if one more non-zero row was present? 5. What is the geometric interpretation of the rank of a matrix? **Tip**: The rank of a matrix represents the maximum number of linearly independent rows (or columns) in the matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Row Echelon Form
Matrix Rank
Formulas
Elementary Row Operations
Rank Definition
Row Echelon Form
Theorems
Rank-Nullity Theorem
Elementary Row Operations
Suitable Grade Level
Grades 11-12, College level