For a `ntimes n` matrix `A=(a_1, cdots, a_n)`, the following statements are equivalent.

• For any vector `y`, there is only one vector `x` such that `y=Ax`.
• `A` is invertible.
• `text{Im}  A` is not suppresed, which means `A` is injective.
• `dim text{ker}  A=0`, which means `text{ker}  A={O}`.
• `a_1`, `cdots`, `a_n` are linearly independent.
• `text{Im}  A` covers the objective space all, which means `A` is surjective.
• `text{rank}  A=dim text{Im}  A=n`.
• `det A ne 0`.
• `A` has not the eigenvalue which is zero.
• So does `A^t`. 

    The following statements are equivalent.

• There exists `y` such that `y ne Ax` for all `x`.
• `A` is not invertible.
• `text{Im}  A` is suppresed, which means `A` is not injective.
  
• `dim text{ker}  A>0`, which means `text{ker}  A` has a element other than `O`.
  
• `a_1`, `cdots`, `a_n` are linearly dependent.
  
• `text{Im}  A` does not cover the objective space all, which means `A` is not surjective.
  
• `text{rank}  A=dim text{Im}  A<n`.
• `det A = 0`.
  
• `A` has the eigenvalue which is zero.
  
• So does `A^t`.
  

︎Reference

[1] Hiraoka Kazuyuki, Hori Gen, Programming No Tame No Senkei Daisu, Ohmsha.