Equivalence theorem for nonsingular matrices: the equation Ax=0 has only the trivial solution.

Created over 8 years ago, updated 10 days ago

Theorem: An $n \times n$ matrix $A$ is nonsingular if and only if $Ax = 0$ has only the trivial solution $x = 0$.

Proof: If $A$ is nonsingular, $x = A^{-1}0 = 0$. Conversely, if only $x = 0$ solves $Ax = 0$, then the null space is $\{0\}$, so nullity is 0, so rank is $n$, so columns are linearly independent, so $A$ is nonsingular.