Math Problem Statement
Let [[1,2,3,5], [2, 4, 8 ,12 ], [3,6,7,13 ]] Then rref(A) =?
Solution
Let's find the Reduced Row Echelon Form (RREF) of the matrix:
1 & 2 & 3 & 5 \\ 2 & 4 & 8 & 12 \\ 3 & 6 & 7 & 13 \end{bmatrix}$$ ### Steps to compute the RREF: 1. **Make the leading entry of the first row 1** (it's already 1, so we leave it unchanged). 2. **Use row operations to create zeros below the first column:** - Subtract 2 times row 1 from row 2: $$R_2 \rightarrow R_2 - 2R_1$$ - Subtract 3 times row 1 from row 3: $$R_3 \rightarrow R_3 - 3R_1$$ After performing these row operations: $$A = \begin{bmatrix} 1 & 2 & 3 & 5 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & -2 & -2 \end{bmatrix}$$ 3. **Make the leading entry in the second row (column 3) 1**: - Divide row 2 by 2: $$R_2 \rightarrow \frac{1}{2} R_2$$ This gives: $$A = \begin{bmatrix} 1 & 2 & 3 & 5 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & -2 & -2 \end{bmatrix}$$ 4. **Use row operations to create zeros above and below the leading 1 in column 3:** - Add 2 times row 2 to row 3: $$R_3 \rightarrow R_3 + 2R_2$$ - Subtract 3 times row 2 from row 1: $$R_1 \rightarrow R_1 - 3R_2$$ After performing these operations, the matrix becomes: $$A = \begin{bmatrix} 1 & 2 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ 5. **Final Check**: The matrix is now in reduced row echelon form (RREF). $$RREF(A) = \begin{bmatrix} 1 & 2 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ ### Summary: The RREF of the given matrix is: $$\begin{bmatrix} 1 & 2 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ --- Would you like more details on any of these steps or explanations about row operations? Here are a few related questions: 1. What does the RREF of a matrix represent geometrically? 2. How can I find the rank of a matrix using RREF? 3. What are the key differences between row echelon form (REF) and RREF? 4. Can all matrices be converted to RREF? 5. How can I use the RREF of a matrix to solve a system of linear equations? **Tip**: RREF is particularly useful in solving systems of linear equations as it reveals the solutions directly when in the form of an augmented matrix!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrices
Row Echelon Form
Reduced Row Echelon Form (RREF)
Formulas
Row Operations
RREF Definition
Theorems
Row Echelon Form (REF) Theorem
Reduced Row Echelon Form Theorem
Suitable Grade Level
Grades 10-12