grandes-ecoles 2023 Q11

grandes-ecoles · France · mines-ponts-maths1__psi 3x3 Matrices Determinant of Parametric or Structured Matrix
Show that, if $A$ and $B$ belong to $S _ { n } ^ { + + } ( \mathrm { R } )$, then:
$$\forall t \in [ 0,1 ] , \quad \operatorname { det } ( ( 1 - t ) A + t B ) \geq \operatorname { det } ( A ) ^ { 1 - t } \operatorname { det } ( B ) ^ { t }$$
Justify that this inequality remains valid for $A$ and $B$ only in $S _ { n } ^ { + } ( \mathbf { R } )$. 
Show that, if $A$ and $B$ belong to $S _ { n } ^ { + + } ( \mathrm { R } )$, then:

$$\forall t \in [ 0,1 ] , \quad \operatorname { det } ( ( 1 - t ) A + t B ) \geq \operatorname { det } ( A ) ^ { 1 - t } \operatorname { det } ( B ) ^ { t }$$

Justify that this inequality remains valid for $A$ and $B$ only in $S _ { n } ^ { + } ( \mathbf { R } )$.
Paper Questions
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