RREF([A|I]) = [I|inv(A)] iff A is invertible

Dependencies:

  1. RREF is unique
  2. Inverse of a matrix
  3. Product of stacked matrices
  4. Row equivalence matrix
  5. Full-rank square matrix is invertible
  6. Full-rank square matrix in RREF is the identity matrix

Let $A$ be an $n$ by $n$ matrix. Then $\operatorname{RREF}([A|I]) = [I|B]$ iff $AB = BA = I$.

Proof

Since $\operatorname{RREF}([A|I])$ is row-equivalent to $[A|I]$, $\operatorname{RREF}([A|I]) = R[A|I] = [RA|R]$.

Since $[RA|R]$ is in RREF, $RA$ is also in RREF.

$A$ is not invertible implies $\operatorname{rank}(A) \neq n$ implies $RA$ has less than $n$ non-zero rows implies $RA \neq I \Rightarrow \operatorname{RREF}([A|I]) \neq [I|B]$ for any $B$.

$A$ is invertible implies $\operatorname{rank}(A) = n$ implies $\operatorname{RREF}(A) = RA = I$ implies $\operatorname{RREF}([A|I]) = [RA|R] = [I|R]$. Since $RA = I$ and $A$ is invertible, $RA = AR = I$.

Dependency for: None

Info:

Transitive dependencies:

  1. /linear-algebra/matrices/gauss-jordan-algo
  2. /linear-algebra/vector-spaces/condition-for-subspace
  3. /sets-and-relations/equivalence-relation
  4. Group
  5. Ring
  6. Polynomial
  7. Field
  8. Vector Space
  9. Linear independence
  10. Span
  11. Integral Domain
  12. Comparing coefficients of a polynomial with disjoint variables
  13. Semiring
  14. Matrix
  15. Stacking
  16. System of linear equations
  17. Product of stacked matrices
  18. Matrix multiplication is associative
  19. Reduced Row Echelon Form (RREF)
  20. Elementary row operation
  21. Every elementary row operation has a unique inverse
  22. Row equivalence of matrices
  23. Matrices over a field form a vector space
  24. Row space
  25. Row equivalent matrices have the same row space
  26. RREF is unique
  27. Identity matrix
  28. Full-rank square matrix in RREF is the identity matrix
  29. Inverse of a matrix
  30. Inverse of product
  31. Elementary row operation is matrix pre-multiplication
  32. Row equivalence matrix
  33. Equations with row equivalent matrices have the same solution set
  34. Rank of a matrix
  35. Basis of a vector space
  36. Linearly independent set is not bigger than a span
  37. Homogeneous linear equations with more variables than equations
  38. Rank of a homogenous system of linear equations
  39. Full-rank square matrix is invertible