Science:Math Exam Resources/Courses/MATH221/December 2011/Question 03 (b)
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Question 03 (b) 

Consider the matrix Compute where . 
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 

Nothing fancy here, just a straight up computation. 
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 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. First, let's compute the matrix B And clearly this matrix has determinant zero. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. Note: this is a more fancy solution, no worries if you don't like it as much. The matrix B is a polynomial in the matrix A that we can factor as And since the determinant is multiplicative, we have that Now in part (a) we saw that 2 is an eigenvalue and hence the determinant of is zero and thus so is the determinant of the matrix B. 