AB = I implies BA = I

Dependencies:

  1. Identity matrix
  2. Rank of a homogenous system of linear equations
  3. Matrix multiplication is associative
  4. Full-rank square matrix is invertible

Let $A$ and $B$ be $n$ by $n$ matrices. If $AB = I$, then $BA = I$.

Proof

$BX = 0$ is a system of $n$ linear equations in $n$ variables.

\[ BX = 0 \implies A(BX) = A0 \implies (AB)X = 0 \implies IX = 0 \Rightarrow X = 0 \] Since $X = 0$ is the only solution to $BX = 0$, $\operatorname{rank}(B) = n$.

Since $\operatorname{rank}(B) = n$, $B$ is invertible. Therefore, every left inverse of $B$ is also a right inverse. Therefore, $BA = I$.

Dependency for:

  1. Orthogonal matrix
  2. Determinant of product is product of determinants

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 homogenous system of linear equations
  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. Full-rank square matrix in RREF is the identity matrix
  39. Full-rank square matrix is invertible