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

Please refer to part (a) regarding the computation of $\parallel A\parallel$

Recall that

${\text{cond}}(A)=\parallel A\parallel \cdot \parallel A^{-1}\parallel$

With A being a diagonal matrix, its inverse is given by:

$A^{-1}={\begin{bmatrix}1&0&0\\0&{\frac {1}{a}}&0\\0&0&{\frac {1}{2}}\end{bmatrix}}$

valid for all values where a is nonzero. When a is zero, this inverse, and hence cond(A) is undefined. Similar to $\parallel A\parallel$ , we have $\parallel A^{-1}\parallel =\max\{\parallel 1\parallel ,\parallel {\frac {1}{a}}\parallel ,\parallel {\frac {1}{2}}\parallel \}$ and these are defined for all nonzer values of a.

So cond(A) is defined for all values of a, except a = 0.