Useful Matrix properties

Theorem

Let $A$ be $m\times n$ matrix, $B$ and $C$ $n\times n$ matrices, with $B$ symmetric, $x$ $n\times 1$ vector

1. $Ax=0 \; \forall x$ if and only if $A=0$,
2. $x^TAx \; \forall x$ if and only if $B=0$,
3. $x^TCx \; \forall x$ if and only if $C^T=-C$ (skew symmetric).