How to find reduced row echelon form?
Understand the Problem
The question is asking how to convert a matrix into its reduced row echelon form (RREF), a form that makes it easier to solve systems of linear equations. To achieve this, one typically applies a sequence of row operations on the matrix.
Answer
The matrix in reduced row echelon form is: $$ RREF(A) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} $$
Answer for screen readers
The final matrix in reduced row echelon form (RREF) will look like this: $$ RREF(A) = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} $$ (example for a typical matrix)
Steps to Solve
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Identify the Matrix Identify the matrix you wish to convert to RREF. It could look something like this: $$ A = \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{pmatrix} $$
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Get Leading 1s Perform row operations to get leading coefficients (the first non-zero number from the left, in a row) to be 1. Typically you can do this by dividing the rows by their leading coefficients.
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Zero Out Below Leading 1s Use row operations to create zeros below each leading 1 in its column. For instance, you can subtract a multiple of the row containing the leading 1 from the rows below it.
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Zero Out Above Leading 1s After you have zeros below each leading 1, go back and create zeros above each leading 1 as well.
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Repeat as Necessary Repeat the process of creating leading 1s and clearing out columns until all rows are in the required form.
The final matrix in reduced row echelon form (RREF) will look like this: $$ RREF(A) = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} $$ (example for a typical matrix)
More Information
The reduced row echelon form is essential for solving systems of linear equations efficiently. Each leading 1 and zero pattern allows one to easily interpret solutions, whether they are unique, infinite, or inconsistent.
Tips
- Not applying row operations correctly, leading to an incorrect RREF.
- Forgetting to eliminate entries above leading 1s, which is crucial for reaching proper RREF.
- Misidentifying the leading coefficients or making errors during arithmetic operations.
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