Math_152_April_2006

Math_152_April_2006 - April 2006 Mathematics 152 Page 2 of...

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Unformatted text preview: April 2006 Mathematics 152 Page 2 of 9 pages Marks [12] 1. [Short answer] (a) [3] Let A = x- y y x and B = s- t t s . Is it true that AB = BA for all choices of x , y , s and t ? (b) [3] Write down a 2 2 matrix with real entries but with complex eigenvalues. (c) [3] For which values of a does 1 a 1 have only one eigenvector (up to scalar multiples)? (d) [3] If A = 1 1 1 what is A 100 ? Continued on page 3 April 2006 Mathematics 152 Page 3 of 9 pages [12] 2. For which values of a and b are the vectors 1 1 1 , - 2 2 and 1 a b linearly independent? Continued on page 4 April 2006 Mathematics 152 Page 4 of 9 pages [14] 3. Let T be the triangle in three dimensional space with vertices located at p = 1 1 1 , q = 1 2 3 and r = 1- 1 (a) [7] What is the cosine of the angle at the vertex p ? (b) [7] What is the area of the triangle. (Hint: it is half the area of the parallelogram spanned by two of its sides.) Continued on page 5 April 2006...
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Math_152_April_2006 - April 2006 Mathematics 152 Page 2 of...

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