Gradient and Hessian of log-determinant

ProofAdvanced⏱ 30 minFoundationslog determinantmatrix derivativeconcavity

Problem

For $X \succ 0$ prove that

\nabla_X \log\det X = X^{-1},

and that $X \mapsto \log\det X$ is concave on the cone of positive definite matrices.

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints — 0 of 3 revealed

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