Let $n \in \mathbb { N } ^ { * }$. (a) Using the results of questions 14 and 15, show that for the matrix $A _ { n }$ defined at the beginning of part 2, we have: $$N \left( \left( ( n + 1 ) ^ { 2 } A _ { n } \right) ^ { - 1 } \right) \leq \frac { 1 } { 8 }$$ (b) Deduce that for any diagonal matrix $D _ { n } = \left[ d _ { i , j } \right] _ { 1 \leq i , j \leq n }$ such that $d _ { i , i } \geq 0$ for all $i \in \{ 1 , \ldots , n \}$, we also have $$N \left( \left( ( n + 1 ) ^ { 2 } A _ { n } + D _ { n } \right) ^ { - 1 } \right) \leq \frac { 1 } { 8 }$$
Let $n \in \mathbb { N } ^ { * }$.
(a) Using the results of questions 14 and 15, show that for the matrix $A _ { n }$ defined at the beginning of part 2, we have:
$$N \left( \left( ( n + 1 ) ^ { 2 } A _ { n } \right) ^ { - 1 } \right) \leq \frac { 1 } { 8 }$$
(b) Deduce that for any diagonal matrix $D _ { n } = \left[ d _ { i , j } \right] _ { 1 \leq i , j \leq n }$ such that $d _ { i , i } \geq 0$ for all $i \in \{ 1 , \ldots , n \}$, we also have
$$N \left( \left( ( n + 1 ) ^ { 2 } A _ { n } + D _ { n } \right) ^ { - 1 } \right) \leq \frac { 1 } { 8 }$$