By applying elementary transformations to a matrix, its rank is: A. Increases B. Decreases C. Does not change D. None of these.
Understand the Problem
The question is asking about the effect of applying elementary transformations to a matrix on its rank, specifically how these transformations affect whether the rank increases, decreases, does not change, or falls into none of these categories.
Answer
C: Does not change.
Answer for screen readers
The rank does not change when applying elementary transformations to a matrix.
Therefore, the correct answer is C: Does not change.
Steps to Solve
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Understanding Elementary Transformations Elementary transformations involve three types:
- Row swapping (interchanging two rows).
- Scaling a row by a non-zero scalar.
- Adding or subtracting multiples of one row to another.
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Effect on Rank The rank of a matrix is defined as the maximum number of linearly independent row or column vectors.
- Row swapping and scaling do not change the set of linearly independent rows.
- Adding a multiple of one row to another can change the rank only if it results in creating a new independent row, which generally does not happen.
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Conclusion Since elementary transformations do not change the linear independence of the rows, the rank of the matrix remains unchanged.
The rank does not change when applying elementary transformations to a matrix.
Therefore, the correct answer is C: Does not change.
More Information
Elementary transformations are used in processes like Gaussian elimination to simplify matrices without altering their rank. This property is crucial for solving systems of linear equations.
Tips
- Thinking Rank Increases: Some may assume that adding rows will always create new independent rows. However, if the new rows can be expressed as combinations of existing rows, the rank will remain unchanged.
- Confusing Transformations: Mixing up the effects of elementary transformations can lead to incorrect conclusions about changes in rank.
