How do you determine the rank of a matrix from its echelon form?
Conditions
- The matrix has been reduced to echelon form.
- Pivots are clearly identifiable.
Reasoning, step by step
- Reduce the matrix to echelon form using row operations.
- Identify the pivot in each nonzero row.
- Count the total number of pivots.
- The count is the rank of the matrix.
Example
The completed example has three pivots, so its rank is three. Echelon form already suffices for counting rank; every entry need not be reduced further once pivots are clear.
Common misconceptions
- Believing that the matrix must be in reduced row echelon form (RREF) to count rank.
- Counting zero rows as contributing to the rank.
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