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