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. For $1 \leqslant i \leqslant k$, let $A _ { i }$ be positive definite symmetric matrices and $\alpha _ { i }$ strictly positive real numbers such that $\alpha _ { 1 } + \cdots + \alpha _ { k } = 1$. Prove that $$\operatorname { det } \left( \alpha _ { 1 } A _ { 1 } + \cdots + \alpha _ { k } A _ { k } \right) \geqslant \left( \operatorname { det } A _ { 1 } \right) ^ { \alpha _ { 1 } } \ldots \left( \operatorname { det } A _ { k } \right) ^ { \alpha _ { k } }$$ One may reason by induction on $k$.
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.
For $1 \leqslant i \leqslant k$, let $A _ { i }$ be positive definite symmetric matrices and $\alpha _ { i }$ strictly positive real numbers such that $\alpha _ { 1 } + \cdots + \alpha _ { k } = 1$. Prove that
$$\operatorname { det } \left( \alpha _ { 1 } A _ { 1 } + \cdots + \alpha _ { k } A _ { k } \right) \geqslant \left( \operatorname { det } A _ { 1 } \right) ^ { \alpha _ { 1 } } \ldots \left( \operatorname { det } A _ { k } \right) ^ { \alpha _ { k } }$$
One may reason by induction on $k$.