grandes-ecoles 2017 Q7

grandes-ecoles · France · x-ens-maths__pc Matrices True/False or Multiple-Select Conceptual Reasoning

Say, by briefly justifying the answer, whether the following assertions are correct for all $A , B \in M _ { n } ( \mathbb { C } ) , \mu \in \mathbb { C }$. i) $\rho ( \mu A ) = | \mu | \rho ( A )$. ii) $\rho ( A + B ) \leqslant \rho ( A ) + \rho ( B )$. iii) $\rho ( A B ) \leqslant \rho ( A ) \rho ( B )$. iv) For $P \in M _ { n } ( \mathbb { C } )$ invertible, $\rho \left( P ^ { - 1 } A P \right) = \rho ( A )$. v) $\rho \left( { } ^ { t } A \right) = \rho ( A )$.

Say, by briefly justifying the answer, whether the following assertions are correct for all $A , B \in M _ { n } ( \mathbb { C } ) , \mu \in \mathbb { C }$.\\
i) $\rho ( \mu A ) = | \mu | \rho ( A )$.\\
ii) $\rho ( A + B ) \leqslant \rho ( A ) + \rho ( B )$.\\
iii) $\rho ( A B ) \leqslant \rho ( A ) \rho ( B )$.\\
iv) For $P \in M _ { n } ( \mathbb { C } )$ invertible, $\rho \left( P ^ { - 1 } A P \right) = \rho ( A )$.\\
v) $\rho \left( { } ^ { t } A \right) = \rho ( A )$.

Paper Questions

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