Mark the following TRUE or FALSE. No explanation needed.

$($a$)||A|{|}_{}=||A^T|{|}_{}$ for any matrix $A.$

$($b$)$ The MATLAB code $A**(B**p)$ runs faster than $(A**B)**p.($Here $A,Bin{R}^{nn},n2,$ and $pin{R}^n.)$

$($c$)$ Positive numbers smaller than ${}_{\text{m}\text{a}\text{c}\text{h}}$ are not representable in a normalized double precision floating point system.

$($d$)$ If the condition number is small enough, then a small backward error implies a small forward error.

$($e$)$ For high dimensional sparse linear systems, a $PA=LU$ decomposition with partial pivoting is often the most efficient algorithm to solve the system.