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I need some assistance in solving this;

Let $ A, B \in M_n$ be positive definite and let $\alpha \in (0,1)$. I need to show that $\alpha A^{-1} + (1-\alpha)B^{-1} \geq ((\alpha A+(1-\alpha)B)^{-1}$ with equality iff A=B. Thus the function $ f(t)=t^{-1}$ is strictly convex on the set of positive definite matrix

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By $\mathbf{X} \geq \mathbf{Y}$ do you mean $\mathbf{X}\succeq\mathbf{Y}$, which implies $\mathbf{X}-\mathbf{Y}$ is positive definite ? If so, then following post answers your question, Is inverse matrix convex?

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    $\begingroup$ Note that the accepted answer in that post is said to have a mistake. $\endgroup$ May 1, 2014 at 22:45
  • $\begingroup$ I think the second answer by @achillehui is correct. $\endgroup$ Jun 5, 2014 at 14:41

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