Math Problem Statement
I have a equation group represented as a matrix, where the first row is -35 35 23 -165, second is 33 -32 -20 163, third is 18 -18 -12 84, and fourth is -73 75 53 -320. How would I convert it to a rref?
Solution
To convert the given system of equations, represented by the matrix:
-35 & 35 & 23 & -165 \\ 33 & -32 & -20 & 163 \\ 18 & -18 & -12 & 84 \\ -73 & 75 & 53 & -320 \end{pmatrix}$$ into **reduced row echelon form (rref)**, you need to perform a series of elementary row operations. The goal is to get a matrix where: 1. The leading entry in each non-zero row is 1 (a **pivot**). 2. The leading entry of each row is the only non-zero entry in its column. 3. Any rows of all zeros are at the bottom. 4. The rows are ordered so that each pivot is to the right of the pivot in the row above. Here's the step-by-step outline of the Gaussian elimination process to achieve the **rref**: ### Step 1: Begin with the original matrix $$\begin{pmatrix} -35 & 35 & 23 & -165 \\ 33 & -32 & -20 & 163 \\ 18 & -18 & -12 & 84 \\ -73 & 75 & 53 & -320 \end{pmatrix}$$ ### Step 2: Make the first element (row 1, column 1) a leading 1 We can divide row 1 by -35: $$\begin{pmatrix} 1 & -1 & -\frac{23}{35} & \frac{165}{35} \\ 33 & -32 & -20 & 163 \\ 18 & -18 & -12 & 84 \\ -73 & 75 & 53 & -320 \end{pmatrix}$$ This simplifies the first row as follows: $$\begin{pmatrix} 1 & -1 & -\frac{23}{35} & \frac{165}{35} \\ 33 & -32 & -20 & 163 \\ 18 & -18 & -12 & 84 \\ -73 & 75 & 53 & -320 \end{pmatrix}$$ ### Step 3: Eliminate the entries below the pivot (row 1, column 1) We perform row operations to zero out the entries below the first pivot (in column 1): - Row 2: $$R_2 \rightarrow R_2 - 33 \times R_1$$ - Row 3: $$R_3 \rightarrow R_3 - 18 \times R_1$$ - Row 4: $$R_4 \rightarrow R_4 + 73 \times R_1$$ ### Step 4: Normalize the second row to get a pivot of 1 After simplifying the rows, you repeat the process for the second column, making the leading entry in the second row equal to 1, and eliminating the entries above and below it. You'll follow similar steps to make the other rows into the reduced row echelon form. Would you like me to continue with the specific calculations for each row, or would you prefer to proceed with the matrix operations on your own? Let me know how you'd like to move forward!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrices
Row Echelon Form
Gaussian Elimination
Formulas
Row operations for Gaussian elimination
Reduced row echelon form (rref) conditions
Theorems
Gaussian elimination
Row echelon form theorem
Reduced row echelon form theorem
Suitable Grade Level
Grades 11-12