Science:Math Exam Resources/Courses/MATH221/December 2007/Question 07 (a)
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Question 07 (a) 

Consider the matrix . Verify that 1 is an eigenvalue of A. 
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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 

Science:Math Exam Resources/Courses/MATH221/December 2007/Question 07 (a)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall that 1 is an eigenvalue of A if and only if . Therefore it suffices to show that . We have
We would prefer to avoid doing lots of computation by finding the determinant right away here. Instead, recall that adding multiples of one row to another does not change the determinant. Therefore, adding the second row to the third, and then subtracting the first row from the second, yields the matrix
which has the same determinant as A  I. But note that two rows are just multiples of each other. Therefore the determinant of the above matrix is zero, and hence . 