The following statements are equivalent for the matrix mapping $L_A$ from $\mathbb{R}^n$ into $\mathbb{R}^n$ determined by the $n\times n$ matrix $A$ :

• $L_A$ is invertible
  
  
• $\ker{L_A}=\{\vec{0}\}$
  
  
• $\text{im}(L_A)=\sbspmatrix{L_A(\vec{e}_1), \ldots, L_A(\vec{e}_n)}=\mathbb{R}^n$
  
  
• $\text{rank}(A)=n$
  
  
• $\det(A)\neq 0$
  
  
• $A$ is invertible