Algonquin College Linear Algebra Worksheet

Consider the following.A-[-2]
1 5
8
(a) Compute the characteristic polynomial of A.
det(A – 21) =
x
(b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (Repeated eigenvalues should be entered repeatedly with the same eigenspaces.)
11
=
has eigenspace span
(smallest i-value)
12 =
has eigenspace span
(largest 2-value)
11
(c) Compute the algebraic and geometric multiplicity of each eigenvalue.
14 has algebraic multiplicity
X and geometric multiplicity 1
12 has algebraic multiplicity
X and geometric multiplicity 1
5
-1
A-
-3
-5.A
a) det (A-XI)
s
–
V
1
-3-1
= (-5-A) (-3-1) +
= 15+54+32+x²
– 42 +81 +15
= (x+4)2
+
Eigenvalues: -n-4
b)
te – 4
Byt 4 of
(A-(-4)I)
1
-3+A
(b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (Repeated eigenvalues should be entered repeatedly with the same eigenspaces.)
LE:
1 =
has eigenspace
span
(smallest 2-value)
12 =
has eigenspace span
in =
has eigenspace span
(largest 2-value)
11