MATH 320 PRACTICE QUESTIONS FOR FINAL
These practice questions cover material since the second midterm. Half of the
actual final will consist of questions on this material, and the other half will be on
older material. Before attempting these problems, review the notion of minimal
polynomial by doing the next two problems from the book.
(1) Do problems 1.13 and 1.14 on page 393.
There are about twice as many questions here as will appear regarding this
material on the final. You are encouraged to take it in exam conditions—no notes,
calculator, book, etc., and keeping time constraints in mind. Solutions will be
posted on Thursday, Dec. 6.
For all problems, you must fully explain your reasoning. Write in complete
sentences when appropriate.
Problem 1. Determine all values of x∈Rsuch that the matrix
A=
4 0 1 3
x x 2 5
x0 0 1
0 0 4 0
is invertible.
Problem 2. Let
A=
2 0 0
13−1
2 2 0
.
(1) Determine all eigenvalues of A.
(2) Determine a basis for each eigenspace of A.
(3) Determine the algebraic and geometric multiplicities of each eigenvalue.
(4) Determine if Ais diagonalizable. If it is, write an equation of the form
D=C−1AC, with Ddiagonal.
Problem 3. Let a1,...,a5be real numbers. Compute
det a11
1 0a21
1 0a31
1 0a41
1 0a51
1 0.
Problem 4. Let
A=
6−4 4
8−6 8
002
.
(1) Compute the characteristic polynomial pA(x).
(2) Compute the minimal polynomial mA(x).
(3) Use the minimal polynomial to compute A100. (Your answer may contain
unsimplified powers of numbers.)
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