Practice Exam #3
Math 2270, Spring 2005
Problem 1. Let A be a 3 × 3 matrix with rows ~v1 , ~v2 , ~v3 and det(A) = 2. Find the following
determinants:


~v1
(a) (4 points) det  4~v2  =
~v1 + ~v3


2~v1 − ~v2
(b) (4 points) det  −4~v1 + 2~v2  =
~v3
(c) (4 points) det(B), where B is the 3 × 3 matrix with colomns ~v1T , ~v2T , ~v3T .


−~v2
(d) (4 points) det  ~v1  =
−3~v3
Problem 2. (10 points) Let T : R2×2 → R2×2 be the linear transformation given by T (A) = −AT .
Find det(T ).


1 a b
Problem 3. Let A =  0 1 c .
0 0 2
(a) (5 points) Find the eigenvalues of A.
(b) (5 points) Find the algebraic multiplicities of the eigenvalues from (a).
(c) (10 points) How the geometric multiplicities of the eigenvalues from (a) depend on the
constants a, b, c?
(d) (5 points) Find a, b, c such that there is an eigenbasis for A.
(e) (5 points) With the values from (c) find an eigenbasis for A.
Problem 4. (10 points) Consider the vectors ~v1 , ~v2 , . .p
. , ~vm in Rn . Then the m-volume of the
m-parallelepiped defined by the vectors ~v1 , ~v2 , . . . , ~vm is det(AT A), where A is the n × n matrix
with columns ~v1 , ~v2 , . . . , ~vm .
Problem 5. Give the definition of the following notions:
(a) (5 points) Rotation matrix. Give an example of a 2 × 2 rotation matrix.
(b) (5 points) Eigenspace. Give an example of an eigenspace.
Problem 6. True or false? Justify all your answers.
(a) (6 points) There exists 3 × 3 invertible matrices A and S such that S −1 AS = −A.
(b) (6 points) If A is a 3 × 3 matrix given by A = [~v1 ~v2 ~v3 ], then | det(A)| = k~v1 k k~v2 k k~v3 k.
(c) (6 points) If two square matrices have the same characteristic polynomial, then they must
be similar.
(d) (6 points) If A is a 3 × 3 matrix and 1 + 2i and 1 are eigenvalues, then A is diagonalizable
over C.