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