grandes-ecoles 2019 Q38

grandes-ecoles · France · centrale-maths1__pc Matrices Determinant and Rank Computation
We fix a $\mathbb{C}$-vector space $E$ of dimension $n \geqslant 2$. Let $\mathcal{A}$ be an irreducible subalgebra of $\mathcal{L}(E)$ (i.e., the only vector subspaces stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$).
Deduce from Q37 the existence of an element of rank 1 in $\mathcal{A}$. 
We fix a $\mathbb{C}$-vector space $E$ of dimension $n \geqslant 2$. Let $\mathcal{A}$ be an irreducible subalgebra of $\mathcal{L}(E)$ (i.e., the only vector subspaces stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$).

Deduce from Q37 the existence of an element of rank 1 in $\mathcal{A}$.
Paper Questions
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