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, $\alpha$ and $\beta$ two real numbers $> 0$ such that $\alpha + \beta = 1$; prove that: $$\operatorname { det } ( \alpha A + \beta B ) \geqslant ( \operatorname { det } A ) ^ { \alpha } ( \operatorname { det } B ) ^ { \beta }$$
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, $\alpha$ and $\beta$ two real numbers $> 0$ such that $\alpha + \beta = 1$; prove that:
$$\operatorname { det } ( \alpha A + \beta B ) \geqslant ( \operatorname { det } A ) ^ { \alpha } ( \operatorname { det } B ) ^ { \beta }$$