#Type/Theorem #Topic/Linear_Algebra

Types: 14.2 The Invertible Matrix Theorem Part 2, 14.3 The Invertible Matrix Theorem Part 3
Examples: N/A
Constructions: N/A
Generalizations: N/A

Properties: N/A
Sufficiencies: N/A
Questions: N/A

The Invertible Matrix Theorem: Part 1

Let A∈Rn×n be a square matrix with real entries. Then, the following statements are equivalent ⟺. In other words, for a given matrix A, the statements are either all true or all false.

  1. There is a matrix C∈Rn×n such that CA=In.
  2. There is a matrix D∈Rn×n such that AD=In.
  3. A is an invertible matrix (A is nonsingular).
  4. A is a row equivalent to an 8.9 Upper-Triangular Matrix with nonzero entries on the main diagonal.
  5. A has n pivot positions.
  6. The matrix equation Ax→=0→ has only the trivial solution x→=0.
  7. The columns of A are linearly independent. In other words,
isalinearlyindependentsetofvectors.8.Thelineartransformation$f(x→)=Ax→$isone−to−one.9.Theequation$Ax→=b→$hasauniquesolutionsforall$b→∈Rn$.10.Thecolumnsof$A$[[LinearAlgebra/06SpanandLinearIndependence/6.3Spanofasetofvectors∥span]]$Rn$.11.Thelineartransformation$f(x→)=Ax→$maps$Rn$onto[1]$Rn$.12.Thematrix$AT$isinvertible.

\begin{proof}
to be added :(
\end{proof}