Science:Math Exam Resources/Courses/MATH307/December 2006/Question Section 102 07 (d)
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Question Section 102 07 (d) 

Decide whether or not the following statement is true or false. If true, give a proof. If false, give a counterexample. (d) For every 5x5 matrix A with real entries, 
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 

Show that the left hand side of the problem is actually the square of a real number. 
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. TRUE. Proof: Note 1:Since A is a square matrix, we know that is true. Note 2: Since A is a square matrix, we also know that and hence
Since the determinant is squared, this means that the number is always positive. i.e. . 