Proving the Even Rank of Skew Symmetric Matrices: Induction and Other Methods

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SUMMARY

The rank of a skew-symmetric matrix is definitively even, as established through induction and rank-nullity theory. The dimension of a skew-symmetric matrix is calculated as n(n-1)/2, where n represents the number of rows or columns. This dimension directly supports the conclusion regarding the even rank. Additionally, the discussion highlights the importance of understanding the null space when calculating dimensions related to the equation AX=0.

PREREQUISITES
  • Understanding of skew-symmetric matrices
  • Familiarity with mathematical induction
  • Knowledge of rank-nullity theorem
  • Basic linear algebra concepts
NEXT STEPS
  • Study the properties of skew-symmetric matrices
  • Explore the rank-nullity theorem in detail
  • Learn about the implications of matrix dimensions on linear transformations
  • Investigate alternative proofs for the even rank of skew-symmetric matrices
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Mathematicians, students studying linear algebra, and researchers interested in matrix theory and its applications.

bernoli123
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how can we prove that the rank of skew symmetric matrix is even
i could prove it by induction
is there another way
 
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what about thinking of rank-nullity theory
since the dimension of this skew-symmetric matrix=n(n-1)/2
but how to calculate the dim of the AX=0
 

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