Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We set $$A = \left( \begin{array} { l l } a & b \\ b & d \end{array} \right)$$ and assume $A \geqslant 0$ (i.e., all eigenvalues of $A$ are $\geqslant 0$). II.D.1) Prove that $\operatorname { det } ( A ) \geqslant 0$. II.D.2) Prove that ${ } ^ { t } X A X \geqslant 0$ for every vector $X$. II.D.3) Prove that $a \geqslant 0$ and $d \geqslant 0$. II.D.4) Let $S \in \mathcal { M } _ { 2 } ( \mathbb { R } )$ be symmetric. Prove that: $$S \geqslant 0 \quad \text { if and only if } \quad ( \operatorname { Tr } ( S ) \geqslant 0 \text { and } \operatorname { det } ( S ) \geqslant 0 )$$
Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We set
$$A = \left( \begin{array} { l l } a & b \\ b & d \end{array} \right)$$
and assume $A \geqslant 0$ (i.e., all eigenvalues of $A$ are $\geqslant 0$).
II.D.1) Prove that $\operatorname { det } ( A ) \geqslant 0$.
II.D.2) Prove that ${ } ^ { t } X A X \geqslant 0$ for every vector $X$.
II.D.3) Prove that $a \geqslant 0$ and $d \geqslant 0$.
II.D.4) Let $S \in \mathcal { M } _ { 2 } ( \mathbb { R } )$ be symmetric. Prove that:
$$S \geqslant 0 \quad \text { if and only if } \quad ( \operatorname { Tr } ( S ) \geqslant 0 \text { and } \operatorname { det } ( S ) \geqslant 0 )$$