Proving that a matrix is an inverse of another.

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SUMMARY

The discussion revolves around proving that if matrices C and D exist such that CA = In and AD = Im for an m x n matrix A, then C must equal D. The participants emphasize the implications of these equations on the properties of matrix A, particularly regarding its pivot rows and columns. The assumption that m ≤ n is critical for the proof, as it allows for the exploration of the relationships between the dimensions of A and its inverse properties.

PREREQUISITES
  • Understanding of matrix multiplication and properties of identity matrices.
  • Familiarity with concepts of pivot rows and columns in linear algebra.
  • Knowledge of the definitions and properties of inverse matrices.
  • Basic understanding of dimensions in matrices (m x n notation).
NEXT STEPS
  • Study the properties of inverse matrices in linear algebra.
  • Learn about the implications of the rank-nullity theorem on matrix dimensions.
  • Explore the concept of pivot positions and their significance in solving linear equations.
  • Investigate the conditions under which a non-square matrix can have a left or right inverse.
USEFUL FOR

This discussion is beneficial for students and educators in linear algebra, particularly those focusing on matrix theory and properties of inverses. It is also useful for anyone preparing for advanced mathematics or engineering courses that involve matrix computations.

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Homework Statement



Let A be an m x n matrix, and suppose there exist n x m matrices C and D such that CA = In and AD = Im. Prove that C = D.

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The Attempt at a Solution



I think it's obvious that C=D=A^(-1). But I'm having trouble proving it since I cannot prove that A is a square matrix and I'm not sure how where to start trying to prove it if A is not square.
 
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Well, WLOG assume that m <= n.

If AD = Im, then what does that say about A? That means for any m-dimensional vector b, there is an n-dim vector x such that Ax = b (since one can choose x = Db). What does that say about the number of pivot rows of A?

If CA = In, that means for any n-dimensional vector b, there is an m-dim vector x such that A^Tx = b (one can choose x = C^Tb). What does that say about the number of pivot rows of A^T? What does that say about the number of pivot columns of A?

Then put those together. :)
 
Well, WLOG assume that m <= n.

If AD = Im, then what does that say about A? That means for any m-dimensional vector b, there is an n-dim vector x such that Ax = b (since one can choose x = Db). What does that say about the number of pivot rows of A?

If CA = In, that means for any n-dimensional vector b, there is an m-dim vector x such that A^Tx = b (one can choose x = C^Tb). What does that say about the number of pivot rows of A^T? What does that say about the number of pivot columns of A?

Then put those together. :)
 

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