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

Let where a is a real number. (c) For what values of a (if any) is cond(A) = 1/2? Give a reason. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Break this problem up into cases when and when . Compute the norms of A and its inverse in each case and try to show this situation is impossible. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall that
Now, since A is diagonal matrix, it is easy to find .
If , since . And since . Thus, Therefore, it is impossible to get . If , since . And since Thus, Hence, again it is impossible to get . Therefore, there is no such value a which gives 