Exercise $6$ Or at a fixed real. In M$($R$),$ we consider the matrix: A $([3-$a$,$ a$-5,$ a$],[-$a$,$ a$-2$ a $][5-5-2]),$ the canonical basis of R$^3$ and f the endomorphism of R$^3,$ We note B$=($vec$($e$1),$vec$($e$2),$vec$($e$3))$ such that A is the matrix of f by relation to base B $.$For what value$($s$)$ of a is matrix A diagonalizable? For what value$($s$)$ of a is matrix A invertible? We set in this question a$=0$ a$)$ Show that the vectors vec $($u$1)(1,1,0)$;vec$($u$2)=(0,0,1)$ and vec$($u$3)(1\mathrm{.}0,1)$ form a base C of R b$)$ Write the passage matrix P from B to C$.$ and calculate P$^-1.$c$)$ Calculate the matrix D of f by relation to the base C$.($d$)$ Calculate A$"$ for all ninN. e$)$ The formula found here above remains true for n$=-1?$ f$)$ Determine the functions d they x$($t$),$ y$($t$),$ z$($t$)$ are differentiable about R system solutions x$\text{'}($t$)=3$x$($t$)-5$y$($t$)$ y $($t$)=-2$y$($t$)$ z $($t$)=5$x$($t$)5$y$($t$)2$z$($t$)$