Invertible Matrix Theorem ( core )

Let A be a square $n\times n$ matrix. Then the following statements are equivalent. Meaning, for a given $A$ ,the statements are either all true or all false.

  1. $A$ is an invertible matrix
  2. $A$ is row equivalent to the $n\times n$ identity matrix
  3. $A$ has $n$ pivot positions
  4. The equation $A\text{x}=0$ has only the trivial solution
  5. The columns of $A$ form a linearly independent set
  6. The linear transformation $\text{x}\to A\text{x}$ is one-to-one
  7. The equation $A\text{x}=b$ has at least one solution for each $b$ in $\R^n$
  8. The columns of $A$ span $\R^n$
  9. The linear transformation $\text{x}\to A\text{x}$ maps $\R^n$ onto $\R^n$
  10. There is an $n\times n$ matrix $C$ such that $CA=I$
  11. There is an $n\times n$ matrix $D$ such that $AD=I$
  12. $A^T$ is an invertible matrix

( Rank, Null and Col space )

  1. The columns of $A$ form a basis of $\R^n$
  2. $\text{Col }A=\R^n$
  3. $\text{rank }A=n$
  4. $\text{dim Nul A}=0$
  5. $\text{Nul A}=\{0\}$