Math Problem Statement
Complete the reduced row echelon form for the matrix shown.
⎡⎣⎢3402−11101∣∣∣∣16−511⎤⎦⎥[3402−11101|16−511]
⎡⎣⎢100a100c1∣∣∣∣bde⎤⎦
Solution
To solve for the complete reduced row echelon form (RREF) of the matrix
3 & 4 & 0 & 2 & -1 & 1 \\ 1 & 1 & 0 & 1 & 6 & -5 \\ 1 & 1 & 0 & 1 & 1 & 1 \end{array}\right]$$ and transform it into the following form: $$\left[\begin{array}{ccc|ccc} 1 & 0 & 0 & a & b & c \\ 0 & 1 & 0 & d & e & f \\ 0 & 0 & 1 & g & h & i \end{array}\right]$$ we will perform row operations to transform the matrix into reduced row echelon form. Let's go through the process step by step. ### Step 1: Create a leading 1 in the top-left corner (row 1, column 1) We can start by dividing row 2 by 1 (since row 2 already has a leading 1 in column 1). $$R_2 = R_2$$ Thus, the matrix becomes: $$\left[\begin{array}{ccc|ccc} 3 & 4 & 0 & 2 & -1 & 1 \\ 1 & 1 & 0 & 1 & 6 & -5 \\ 1 & 1 & 0 & 1 & 1 & 1 \end{array}\right]$$ ### Step 2: Eliminate non-zero entries in column 1 for rows 1 and 3 We can subtract appropriate multiples of row 2 from rows 1 and 3 to eliminate the entries in column 1. - $$R_1 \leftarrow R_1 - 3R_2$$ - $$R_3 \leftarrow R_3 - R_2$$ After these operations, the matrix becomes: $$\left[\begin{array}{ccc|ccc} 0 & 1 & 0 & -1 & 19 & -14 \\ 1 & 1 & 0 & 1 & 6 & -5 \\ 0 & 0 & 0 & 0 & -5 & 6 \end{array}\right]$$ ### Step 3: Create a leading 1 in the second column (row 2, column 2) We already have a leading 1 in row 2, column 2. Now, we just need to subtract row 2 from row 1 to eliminate the entry in row 1, column 2. - $$R_1 \leftarrow R_1 - R_2$$ This gives us the following matrix: $$\left[\begin{array}{ccc|ccc} 0 & 0 & 0 & -2 & 13 & -9 \\ 1 & 1 & 0 & 1 & 6 & -5 \\ 0 & 0 & 0 & 0 & -5 & 6 \end{array}\right]$$ ### Step 4: Handle the third row It appears that we have a zero row in the third row, so we need to move forward with other operations. Thus, the reduced row echelon form of the matrix is: $$\left[\begin{array}{ccc|ccc} 0 & 0 & 0 & -2 & 13 & -9 \\ 1 & 1 & 0 & 1 & 6 & -5 \\ 0 & 0 & 0 & 0 & -5 & 6 \end{array}\right]$$ Let me know if you'd like more detailed explanations or have questions. ### 5 Related Questions: 1. What are the conditions for a matrix to be in reduced row echelon form? 2. How do row operations affect the solution of a system of equations? 3. What does it mean when a row of all zeros appears in a matrix during row reduction? 4. Can every matrix be transformed into reduced row echelon form? Why? 5. How does reduced row echelon form relate to solving linear systems? ### Tip: When reducing matrices, keep track of row operations carefully to avoid mistakes—writing them down step-by-step can help ensure accuracy.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Reduced Row Echelon Form
Formulas
Row reduction operations: swap, multiply, and add rows
Reduced row echelon form conditions
Theorems
Elementary Row Operations
Gaussian Elimination
Rank-Nullity Theorem
Suitable Grade Level
Grades 10-12 or College Level