MTH 221 Practice Final Exam Page 1 of 3
Prof. Kim April 27, 2021 Name_________________________________________

1. Consider a matrix
(1) Find all eigenvalues of .

(2) A vector is an eigenvector of corresponding to one of the eigenvalues of .

2. Consider a matrix . A vector is an eigenvector of corresponding to
eigenvalue . What is ? (Your answer must be a simplified vector)

3. Let be an eigenvector of corresponding to an eigenvalue .

Show that becomes an eigenvector of as well.

4. Consider the matrix

Find all eigenvalues of

5. Consider the matrix . Determine whether is diagonalizable.

Your answer must be YES or NO with justification (no credit without justification).

A = 7 4
−3 −1

⎥.
A
!v = 2
−3

⎥ A A
A5!v
B =
2 0 0 0
0 2 0 0
1 0 −1 0
−1 1 0 −1
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!v =
0
3
0
1
!
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&
&
&
&
B
2 B3!v
!v A25 4
!v 12 A + 6I25
A=
4 0 −60 0 01 2 −3

.
A.
A =
5 0 0 0
0 5 0 0
1 4 −3 0
−1 −2 0 −3
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A
MTH 221 Practice Final Exam Page 2 of 3
Prof. Kim April 27, 2021 Name_________________________________________

6. Let Analyze the long term behavior of the dynamical system defined by
with

7. Let and .
(1) Find , the orthogonal projection of onto .

(2) Write as the sum of two orthogonal vectors, one in and one orthogonal to .

8. Orthogonally diagonalize a symmetric matrix

9. Consider a set , where
(1) Show that is an orthogonal set.

(2) The set is an orthogonal basis for . Express the vector

as the linear combination of the
vectors in

10. Consider a set , where

(1) Show that the set forms an orthonormal basis for .

A = 0.6 0.30.4 0.7
!
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&.
!xk+1 = A
!xk k = 0,1,2,…( ), !x0 = 0.50.5
!
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!y = 63

!v = 3
−1

proj!v
!y
!y
!v

!y Span
!v{ } !v
A = 3 44 9

⎥.
S =
!v1,
!v2 ,
!v3{ }
!v1 =
3
0
3

, !v2 =
1
−4
−1

, !v3 =
2
1
−2

.
S
S !3
!x =
1
1
1

S.
S = !u1 ,!u2 ,!u3{ } !u1 =
1 103 203 20

,!u2 =
3 10
−1 20
−1 20

,!u3 = 0−1 21 2

.
S !3
MTH 221 Practice Final Exam Page 3 of 3
Prof. Kim April 27, 2021 Name_________________________________________

(2) Express the vector

as the linear combination of the vectors in

11. Let .

(1). Show that is an orthogonal basis for

(2). Write as the sum of a vector in and a vector orthogonal to .

(3). Find an orthogonal basis of which includes .

(4). Use the info from (3), find an orthonormal basis of which includes the unit vectors in the same
directions of .

12. Find a least-squares solution of for , , .

13. Find a least-squares solution of for , , .

14. Consider the matrix

Since is a symmetric matrix, is orthogonally diagonalizable.
Demonstrate is orthogonally diagonalizable by finding an orthogonal matrix and a diagonal matrix
such that , with .
Use the fact that are all three eigenvalues of (Don’t need to calculate these given eigenvalues).

!x =
3
2
1

S.
!v1 =
3
−1
2

, !v2 =
1
−1
−2

, !y =
−1
2
7

changed from
−1
2
6

⎜⎜

⎟⎟

!v1,
!v2{ } W = Span
!v1,
!v2{ }.

!y W W
!3
!v1&
!v2
!3!v1&
!v2
A!x =
!
b A =
−1 2
2 −3
−1 3

!x = x1x2

!
b =
4
1
2

A!x =
!
b A =
1 1
1 0
0 1
−1 1

!x = x1x2

!
b =
2
5
6
6

A =
6 −2 −1
−2 6 −1
−1 −1 5

.
A A
A Q D
A =QDQT Q−1 = QT
8,6,3 A  