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SIGNATURE Simon Fraser University Department of Mathematics Midterm Examination 2 MATH 232 14 November 2005 11:30–12:
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- The duration of this exam is 50 minutes.
- DO NOT OPEN this test booklet until told to do so.
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- Do ALL your work in this test booklet. You may use the backside of each page for scrap work.
- The value of each question is shown on the left mar- gins.
Question Score Maximum
1 8
2 8
3 8
4 4
5 12
Total 40
- Let A =
and suppose a row echelon form of^ A^ is
B =
[4] (a) Determine a basis for Row(A).
[4] (b) Determine a basis for Row(At) which consists of rows of At.
- Let
A(λ) =
−λ 1 0 0 0 −λ 1 0 0 0 −λ 1 0 0 1 −λ
[4] (a) Show that the determinant of A(λ) is λ^2 (λ^2 − 1).
[4] (b) Determine a basis for NulA(−1).
[2] 4. (a) Using the answers from Question 3 , determine the characteristic poly- nomial of
B =
[2] (b) Using the answers from Question 3, determine a basis for the eigenspace of B corresponding to the eigenvalue − 1.
