grandes-ecoles 2021 Q26

grandes-ecoles · France · centrale-maths1__pc Matrices Determinant and Rank Computation

For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ and let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system. Deduce that $\operatorname { det } G _ { n } = \prod _ { i = 0 } ^ { n } \left\| V _ { i } \right\| ^ { 2 }$.

For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ and let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system. Deduce that $\operatorname { det } G _ { n } = \prod _ { i = 0 } ^ { n } \left\| V _ { i } \right\| ^ { 2 }$.

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