[Math] Inverse of a matrix and its null space

linear algebramatrices

If a matrix $A$ is invertible, then the null space of A, $N(A)$ has only trivial solution.$$$$
1. Is the inverse also true, i.e., if $N(A)$ has only trivial solution, then $A$ is invertible?$$$$
2. If $N(A)$ has non-trivial solution i.e., $N(A) \ne 0$, does it mean that the matrix A is singular (non invertible)? But I read on wikipedia that $A$ is non invertible if and only if $det$ $A=0$, it doesn't say anything about $N(A)\ne0$.

Best Answer

  1. Yes, the Rank-Nullity Theorem tells us if the null space has dimension zero, then the matrix has full rank. If you want to understand it better, it may be helpful to look into its proof. https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem
  2. Similarly, if the null space contains a non-zero vector, then the matrix is singular. (Of course, assuming it is a square matrix.)
Related Question