Understanding Matrix Rank
The rank of a matrix is a fundamental concept in linear algebra that measures the maximum number of linearly independent row or column vectors in the matrix. For a non-zero matrix, the rank provides valuable insights into its properties.
Key Points about Matrix Rank:
- Definition of Rank: The rank of a matrix is defined as the dimension of the vector space spanned by its rows or columns. It determines the solution of systems of linear equations associated with the matrix.
- Non-Zero Matrix: A non-zero matrix is one that contains at least one non-zero element. This implies that there is at least one row or column vector that is not a linear combination of others.
- Minimum Rank: The minimum rank of any non-zero matrix is 1. This is because even if a matrix consists of only one non-zero row or column, that single vector can span a one-dimensional space.
- Implications:
- If a matrix has a rank of 1, it indicates that all rows or columns are scalar multiples of each other.
- If the matrix has a rank greater than 1, it means there are at least two linearly independent vectors present.
Conclusion
Given that a non-zero matrix must have at least one non-zero entry, its rank cannot be 0. Therefore, the rank is always:
- Greater than or equal to 1: Since the minimum rank is 1 for a non-zero matrix, the correct answer is option 'A' (always greater than or equal to 1).
This understanding of rank is crucial, especially in examinations like JEE, as it forms the basis for various applications in linear transformations and systems of equations.