Math Problem Statement
0 −4 2 2 1 0 0 5 −3 −1 1 −4 2 4 1 0 8 2 2 7 0 4 −3 0 2 do ref
Solution
To find the row echelon form (REF) of the given matrix, we will perform Gaussian elimination. The goal is to get zeros below the leading entry in each row while moving from the top-left corner downwards and from left to right across each row.
Given matrix:
0 & -4 & 2 & 2 & 1 \\ 0 & 0 & 5 & -3 & -1 \\ 1 & -4 & 2 & 4 & 1 \\ 0 & 8 & 2 & 2 & 7 \\ 0 & 4 & -3 & 0 & 2 \\ \end{pmatrix}$$ ### Steps: 1. **Swap Rows if Necessary:** Since the first row starts with zero, we'll swap it with the third row to get a leading 1 in the first position. $$\begin{pmatrix} 1 & -4 & 2 & 4 & 1 \\ 0 & 0 & 5 & -3 & -1 \\ 0 & -4 & 2 & 2 & 1 \\ 0 & 8 & 2 & 2 & 7 \\ 0 & 4 & -3 & 0 & 2 \\ \end{pmatrix}$$ 2. **Eliminate First Column Entries Below the Leading 1:** Since all entries below the leading 1 in the first column are already zero, no further row operations are needed for the first column. 3. **Make Leading Entry in Row 2:** The next leading entry should ideally be in the second row, second column. However, since row 2 has a zero in this position, we move to row 3. We can make the entry in row 3, column 2 a leading entry by dividing it by -4. 4. **Continue Row Operations for Remaining Rows:** Proceed with Gaussian elimination to clear out entries below each leading entry, working from top to bottom and left to right. I'll complete these steps and present the resulting REF matrix.The row echelon form (REF) of the given matrix is: $$\begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{pmatrix}$$ This is the identity matrix, indicating that the original matrix is row-equivalent to the identity matrix, thus making it invertible. Would you like more details on the specific steps for each row operation? Here are some related questions you might find interesting: 1. What are the steps to go from REF to reduced row echelon form (RREF)? 2. How does Gaussian elimination determine the invertibility of a matrix? 3. What does it imply if a matrix's REF form has a row of all zeros? 4. How does the REF process differ for matrices with dependent rows? 5. Can all square matrices be transformed to the identity matrix using REF? **Tip:** For consistent row operations, focus on clearing entries column by column, ensuring each leading entry is 1 where possible.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Row Echelon Form (REF)
Formulas
Gaussian elimination process
Row echelon form
Theorems
Row equivalence
Invertibility of a matrix
Suitable Grade Level
Grades 11-12