If every minor of order r of a matrix A is zero, then rank of A is
Understand the Problem
The question is asking what the rank of a matrix A is when every minor of order r is zero, exploring the relationship between minors and matrix rank.
Answer
The rank of $A$ is less than $r$.
Answer for screen readers
The rank of $A$ is less than $r$.
Steps to Solve
- Understanding the rank and minors relationship
The rank of a matrix is defined as the maximum order of its non-zero minors. In this case, we know that every minor of order $r$ is zero.
- Implication of zero minors
If every minor of order $r$ is zero, it indicates that there are no $r \times r$ submatrices with a non-zero determinant. This directly suggests that the rank of the matrix must be less than $r$.
- Conclusion about the rank
Since the rank is defined as the maximum size of the non-zero minors, and all minors of order $r$ are zero, we conclude:
$$ \text{Rank}(A) < r $$
The rank of $A$ is less than $r$.
More Information
This result is a fundamental theorem in linear algebra. It shows that the presence of zero minors indicates a limitation in the linear independence of the rows or columns associated with those minors, confirming that the rank cannot reach $r$ when all such minors are zero.
Tips
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