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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!

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

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