# prove that positive definite matrix is invertible

The question is to prove that positive definite matrix is invertible. I've proved that positive definite matrix has null space of zero vector. So actually we can prove that a matrix has an null space of zero vector is invertible. How to prove this? I need a basic way of proof without using other conclusions (like all eigenvalues are positive so it's invertible or a positive definite matrix is nonsingular so its inverse exists) used in the proving process. Thank you!

• The null space can never be empty because $0$ is always there. – Git Gud Feb 28 '13 at 22:59

Now that you know that $A\in M_n$ it is injective, use the rank-nullity theorem to deduce that $$\mbox{rank}A=n-\mbox{null}A=n.$$ It follows that $A$ is also surjective, hence invertible.