In this part all matrices are of format $(n, n)$, where $n$ is an integer greater than or equal to 2. We say that a real symmetric matrix is positive definite if and only if all its eigenvalues are strictly positive. Let $A$ and $B$ be two positive definite symmetric matrices. III.C.1) Prove that: $\operatorname { det } \left( I _ { n } + B \right) \geqslant 1 + \operatorname { det } B$. III.C.2) Deduce that: $\operatorname { det } ( A + B ) \geqslant \operatorname { det } A + \operatorname { det } B$.
In this part all matrices are of format $(n, n)$, where $n$ is an integer greater than or equal to 2. We say that a real symmetric matrix is positive definite if and only if all its eigenvalues are strictly positive.
Let $A$ and $B$ be two positive definite symmetric matrices.
III.C.1) Prove that: $\operatorname { det } \left( I _ { n } + B \right) \geqslant 1 + \operatorname { det } B$.
III.C.2) Deduce that: $\operatorname { det } ( A + B ) \geqslant \operatorname { det } A + \operatorname { det } B$.