Math Problem Statement
I have a matrix with a first row -35 35 23 -165, second row 33 -32 -20 163, third row 18 -18 -12 84 fourth row -73 75 53 -320. how would i show it in rref form?
Solution
To find the Reduced Row Echelon Form (RREF) of the given matrix, we'll use row operations to simplify it to its RREF.
You provided the matrix as:
-35 & 35 & 23 & -165 \\ 33 & -32 & -20 & 163 \\ 18 & -18 & -12 & 84 \\ -73 & 75 & 53 & -320 \end{pmatrix}$$ ### Steps to convert it into RREF: 1. **Make the leading entry of the first row a 1**: This is done by dividing the entire row by -35 (the first entry in row 1). 2. **Make entries below and above the leading 1 in the first column 0**: Perform row operations to eliminate the entries below and above the pivot (which will be 1 after the first step). 3. **Move to the second column, second row**: Repeat the process by making the pivot in row 2 (second column) equal to 1 and eliminating the other entries in that column. 4. **Continue for the third and fourth columns**: Proceed row by row, column by column, making the pivots 1 and eliminating the rest of the column. 5. **Final check**: Ensure all pivots are 1, and all other entries in the pivot columns are 0. Alternatively, in practice, this process is commonly done using software tools like **R** or **Python**. Below is how you would approach this in **R**: ### R Code: ```R # Define the matrix A <- matrix(c(-35, 35, 23, -165, 33, -32, -20, 163, 18, -18, -12, 84, -73, 75, 53, -320), nrow = 4, byrow = TRUE) # Use the rref function from the 'MASS' package or 'pracma' package library(pracma) rref(A) ``` This will output the RREF of the matrix. Would you like me to walk you through the row operations step-by-step in more detail, or would you like to run this in your environment?Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Row Reduction
Reduced Row Echelon Form (RREF)
Formulas
Row reduction to RREF: using row operations to achieve pivots and zeros
Theorems
Gaussian Elimination
Suitable Grade Level
Undergraduate Level or Advanced High School