Exam # 3
Spring 2005
MATH 2270-01
Instructor: Oana Veliche
Time: 50 minutes
NAME:
ID#:
INSTRUCTIONS
(1) Fill in your name and your student ID number.
(2) Justify all you answers. Correct answers with no justification will not be given any credit.
(3) No books, notes or calculators may be used.
Page #
2
3
4
5
6
7
8
9
Total
Max. # points
10
10
20
12
10
10
13
15
100
# Points
1
2
Problem 1.Let A be a 3 × 3 matrix with rows ~v1 , ~v2 , ~v3 . If det(A) = 3, find the following determinants:


~v1 + ~v2
(a) (5 points) det  4~v2  =
−~v1


2~v2
(b) (5 points) det  3~v3  =
−~v1
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Problem 2. (10 points) Let T : R2×2 → R2×2 be the linear transformation given by
T (A) = A + AT .
Find det(T ).
4
Problem 3. (a) (5 points) Find the eigenvalues of the matrix


2 1 0
A =  0 3 a .
0 0 3
(b) (5 points) Find the algebraic multiplicities of the eigenvalues from (a).
(c) (10 points) How do the geometric multiplicities of the eigenvalues from (a) depend on the constant
a?
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(d) (6 points) Find a such that there exists an eigenbasis for A.
(e) (6 points) With the value(s) of a from (d) find an eigenbasis for A.
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Problem 4. (a) (5 points) Give the definition of the m-parallelepiped in Rn .
(b) (5 points) Give the definition of the trace of a square matrix.
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Problem 5. (10 points) Prove that if A is an n × n matrix, then det(A) = det(AT ).
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Problem 6. True or false? Justify all your answers (give a proof when it is true, or give an example in
case it is not true).
(a) (6 points) There exists a 2 × 2 matrix A with the determinant negative and with (real) eigenvalues
λ1 = λ2 .
(b) (7 points) If an invertible matrix A is diagonalizable, then A−1 is also diagonalizable.
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(c) (7 points) If A is a 2 × 2 invertible matrix, then det(adj(A)) = det(A).
(d) (8 points) The matrix A =
0 −a
a 0
is diagonalizable over C for any real constant a.