Full-rank square matrix is invertible

Dependencies:

  1. Rank of a matrix
  2. RREF is unique
  3. Inverse of a matrix
  4. Rank of a homogenous system of linear equations
  5. Matrix multiplication is associative
  6. Row equivalence matrix
  7. Full-rank square matrix in RREF is the identity matrix

Let $A$ be an $n$ by $n$ matrix. Then $\operatorname{rank}(A) = n$ iff $A$ has an inverse.

Proof of 'if' part

Let $A$ have an inverse $B$. Then $AB = BA = I_n$.

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

\[ AX = 0 \implies B(AX) = B0 \implies (BA)X = 0 \implies I_nX = 0 \implies X = 0 \] Therefore, $X = 0$ is the only solution to $AX = 0$. Therefore, $\operatorname{rank}(A) = n$.

Proof of 'only-if' part

Let $\operatorname{rank}(A) = n$.

Since $\operatorname{RREF}(A)$ is row-equivalent to $A$, $\operatorname{RREF}(A) = RA$, where $R$ is an invertible matrix.

Since $\operatorname{rank}(A) = n$, all rows of $\operatorname{RREF}(A) = RA$ are non-zero. Since $RA$ is a square matrix in RREF, $RA = I_n$. Since $R$ is invertible, all left inverses of $R$ are the same as all right inverses of $R$. Since $A$ is a right inverse of $R$, it is also a left inverse. Therefore, $AR = I_n = RA$, so $A$ is invertible.

Dependency for:

  1. AB = I implies BA = I
  2. RREF([A|I]) = [I|inv(A)] iff A is invertible
  3. Determinant of product is product of determinants
  4. A is diagonalizable iff there are n linearly independent eigenvectors
  5. General multivariate normal distribution

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