Linear Algebra HW
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chapter_6.pdf
Hussam Malibari Heckman MAT 242 Spring 2017 Assignment Chapter 6 due 04/04/2017 at 11:59pm MST
In some of the problems in this chapter, you will be asked to enter a basis for a subspace. You should do this by placing the entries of each vector inside of brackets, and giving a list of these vectors, separated by commas. For instance, if your basis is
12
3
, 11
1
,
then you would enter [1,2,3],[1,1,1] into the answer blank.
1. (1 point) Find the characteristic polynomial of the matrix[ 10 −6 1 10
] . (Use x instead of λ.)
p(x) = .
2. (1 point) Find the characteristic polynomial of the matrix −1 −1 00 1 −3
−5 5 0
. (Use x instead of λ.)
p(x) = .
3. (1 point) The eigenvalues of
8 −16 160−2 4 −88
0 0 −12
are
. (Enter your answer as a list of numbers; for example, 1, 2, 3.)
4. (1 point) The eigenvalues of
−3 −4 4 −1 0 3 −5 1 0 0 4 3 0 0 0 −2
are
. (Enter your answer as a list of numbers; for example, 1, 2, 3, 3. If an eigenvalue is repeated, then it should be listed as many times as appropriate.)
5. (1 point) 3 The matrix A =
−3 0 0 0 0 −3 −3 −3 3 0 2 2 −3 0 −2 −2
has
two distinct eigenvalues λ1 < λ2. Find the eigenvalues and a basis for each eigenspace.
λ1 = , whose eigenspace has a basis of . λ2 = , whose eigenspace has a basis of .
6. (1 point) The matrix A =
0 0 05 5 0
5 5 0
has two real
eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and a basis of each eigenspace. λ1 = has multiplicity 1, with a basis of . λ2 = has multiplicity 2, with a basis of .
7. (1 point) Let A = [
−2 2 0 −1
] . Find an invertible matrix
P and a diagonal matrix D such that A = PDP−1.
P = [ ]
, D = [ ]
,
8. (1 point) The matrix C =
37 0 −8412 −5 −24
18 0 −41
has two
distinct eigenvalues, λ1 < λ2: λ1 = has multiplicity . The dimension of the corresponding eigenspace is . λ2 = has multiplicity . The dimension of the corresponding eigenspace is . Is the matrix C diagonalizable? (enter YES or NO)
9. (1 point) If n is a positive integer, then [
2 −24 −4 −2
]n is[ ]
(Hint: Diagonalize the matrix [
2 −24 −4 −2
] first. Note that
your answer will be a formula that involves n. Be careful with parentheses.)
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