Invertible Matrix Theroem

If $A$ is an $n\times n$ invertible matrix then the following are equivalent:

1. $A$ is row reducible to the identity
2. $A$ has $n$ pivot positions, aka $\textrm{rank}(A)=0$
3. The image of $A$ has dimension $n$
   1. The span of the column vectors of $A$ span all of $\R^n$
4. The columns of $A$ are linearly independent
   1. They form a basis of $\R^n$
5. The equation $A\bold x=\bold b$ has a unique solution for $\bold b$
6. The Linear Transformation T: $\R^n\rightarrow\R^n~~ \bold x\rightarrow A\bold x$ maps $\R^n$ onto $\R^n$
   1. This linear transformation is one to one
7. $A^T$ is invertible
8. $\ker(A)=\{\bold 0\}$ the nullspace is just the 0 vector
9. $\det A\ne0$
10. $\dim(\ker(A)) = 0$
11. The only solution to the linear relation of the column vectors of $A$ is the trivial solution

Systems of Linear Equations

$$ A\bold x =\bold b $$

• $A$ is $n \times n$
• if $\bold b = 0$ the system is homogenous, otherwise it is nonhomogenous
• if $\det A\ne0$ the matrix is nonsingular and there is a unique solution to the system and $A^{-1}$ exists
  • $\bold x=\bold A^{-1}\bold b$
  • For a homogenous system ($\bold b = 0$), the only solution is the trivial solution $\bold x = \bold 0$
• if $A$ is singular, $\det A=0$, then solutions either do not exist or do exist but are not unique
  • There is a linearly dependent column vector
  • $A^{-1}$ does not exist
  • For the homogenous system there are infinitely many non-zero solutions in addition to trivial solutions
  • For nonhomogenous system, the system has no solution unless $(\bold b,\bold y)= 0$
    • For all vectors $\bold y$ satisfying $A^\bold y=\bold 0$ where $A^$ is the adjoint of $A$
    • If this condition is met system has infinitely many solutions and the solutions are of the form: $\bold x = \bold x_p+\bold x_c$
      • $\bold x_p$ is paritcular solution to the system
      • $\bold x_c$ is the most general solution of the homogenous system