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

MATH307 December 2010

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Question 01 (b)
Let $A={\begin{bmatrix}1&0&0\\0&a&0\\0&0&2\end{bmatrix}}$ where a is a real number. (b) For what values of a (if any) is cond(A) not defined? Give a reason.

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Hint
Recall that ${\text{cond}}(A)=\parallel A\parallel \cdot \parallel A^{-1}\parallel$. For which values of a is A not invertible?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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.

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