Science:Math Exam Resources/Courses/MATH307/December 2010/Question 01 (d)/Solution 1

For convenience, let's recall our results from part (b) and part(c):

${\begin{aligned}\mathrm {cond} (A)&=\|A\|\|A^{-1}\|=\max\{1,|a|,2\}\max\{1,1/|a|,1/2\}\\&=\max\{|a|,2\}\max\{1,1/|a|\}\end{aligned}}$

and cond(A) is not defined when a = 0. This suggests considering three cases for $\displaystyle |a|$:

• Case 1: If $0<|a|\leq 1$ then

$\displaystyle \mathrm {cond} (A)=(2)(1/|a|)=2/|a|$

• Case 2: If $1\leq |a|\leq 2$ then

$\displaystyle \mathrm {cond} (A)=(2)(1)=2$

• Case 3: If $2\leq |a|$ then

$\displaystyle \mathrm {cond} (A)=|a|(1)=|a|$

So, the sketch looks like this: