Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We set $$A = \left( \begin{array} { l l } a _ { 1 } & b _ { 1 } \\ b _ { 1 } & d _ { 1 } \end{array} \right) \quad B = \left( \begin{array} { l l } a _ { 2 } & b _ { 2 } \\ b _ { 2 } & d _ { 2 } \end{array} \right)$$ We assume in this section that $A \geqslant 0$ and $B \geqslant 0$. II.E.1) By applying the Cauchy-Schwarz inequality to the vectors $( b _ { 1 } , \sqrt { \operatorname { det } A }$ ) and $( b _ { 2 } , \sqrt { \operatorname { det } B }$ ), prove that $$b _ { 1 } b _ { 2 } \leqslant \sqrt { a _ { 1 } a _ { 2 } d _ { 1 } d _ { 2 } } - \sqrt { \operatorname { det } A \operatorname { det } B }$$ II.E.2) By computing $\operatorname { det } ( A + B ) - \operatorname { det } A - \operatorname { det } B$, deduce that $$\operatorname { det } ( A + B ) \geqslant \operatorname { det } ( A ) + \operatorname { det } ( B ) + 2 \sqrt { \operatorname { det } ( A ) \operatorname { det } ( B ) }$$
Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We set
$$A = \left( \begin{array} { l l } a _ { 1 } & b _ { 1 } \\ b _ { 1 } & d _ { 1 } \end{array} \right) \quad B = \left( \begin{array} { l l } a _ { 2 } & b _ { 2 } \\ b _ { 2 } & d _ { 2 } \end{array} \right)$$
We assume in this section that $A \geqslant 0$ and $B \geqslant 0$.
II.E.1) By applying the Cauchy-Schwarz inequality to the vectors $( b _ { 1 } , \sqrt { \operatorname { det } A }$ ) and $( b _ { 2 } , \sqrt { \operatorname { det } B }$ ), prove that
$$b _ { 1 } b _ { 2 } \leqslant \sqrt { a _ { 1 } a _ { 2 } d _ { 1 } d _ { 2 } } - \sqrt { \operatorname { det } A \operatorname { det } B }$$
II.E.2) By computing $\operatorname { det } ( A + B ) - \operatorname { det } A - \operatorname { det } B$, deduce that
$$\operatorname { det } ( A + B ) \geqslant \operatorname { det } ( A ) + \operatorname { det } ( B ) + 2 \sqrt { \operatorname { det } ( A ) \operatorname { det } ( B ) }$$