True or False? For each of the following statements prove the truth or demonstrate the falsity by a counter$-$example, where $Ain{R}^{nn},xin{R}^n.($a$)x^{T}Ax=x^T(\frac{A+A^T}{2})x.($b$)A^k=0$ for some positive integer $k$ implies $A=0.($c$)$ If $A$ is orthogonal then $Ax=b$ can be solved in $O(n^2)$ flops. $($d$)$ If $A$ is symmetric positive definite, and $x$ is nonsingular, then $x^{T}Ax$ is symmetric positive definite. $($e$)$ Let $A=$I,$||*|{|}_F^2=tr(A^{**}A)$ and