Math Problem Statement
Find the RREF of the matrix
⎡⎣⎢⎢⎢⎢⎢⎢−2−2−232−421−4−422−1−2−1⎤⎦⎥⎥⎥⎥⎥⎥
Solution
The Reduced Row Echelon Form (RREF) of the given matrix is:
1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ This shows that the matrix has full rank (rank = 3), and the system of equations represented by this matrix has a unique solution. Would you like more details or have any questions? Here are five related questions for deeper exploration: 1. What is the significance of the rank of a matrix? 2. How does RREF relate to solving systems of linear equations? 3. Can a matrix always be reduced to RREF? 4. What are the conditions for a matrix to have a unique solution based on RREF? 5. How does RREF help in determining the linear independence of vectors? **Tip:** The number of non-zero rows in the RREF of a matrix corresponds to the rank of the matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Reduced Row Echelon Form (RREF)
Formulas
Elementary row operations: row swapping, row scaling, and row addition
Theorems
Rank-Nullity Theorem
Existence and Uniqueness Theorem for Linear Systems
Suitable Grade Level
Undergraduate - Linear Algebra