Do elementary row operations preserve the rank of a matrix?
Conditions
- The operations are elementary row operations (swapping rows, scaling rows, adding multiples of one row to another).
Reasoning, step by step
- Apply elementary row operations to the matrix.
- Observe that the row space remains unchanged.
- Reduce the matrix to echelon form.
- Identify the pivot rows in the echelon form.
- Count the pivots to determine the rank.
Example
Elementary row operations preserve rank. Reducing toward echelon form reveals which rows contain pivots and which become zero.
Common misconceptions
- Thinking that row operations change the rank of the matrix.
- Believing that only column operations preserve rank.
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