Final Exam for M229 Matrices and Linear Equations
Name:
1 point to lose here!
9
11
12
Bates
Elkin
Student number:
M229 Matrices and Linear Equations
Final Exam
Version SAMPLE


1 −2 −4 0 7
3
7 1 11 .
1. A linear system Ax = b has augmented matrix  −2
0
1
1 0 4
(8 pts) A) Write the vector form of the general
solution as a linear combination of the distinguished solution and basic solutions.
(2 pts) B) What is the rank of A?
Y
1 P
(6 pts) 2. A linear function fA has the indicated
transform plot.
Find the matrix A and compute its determinant.
Y
fA
11
00
00
11
1
1
0
Q’
Q
−1
1
X
1
−1
1
0
0
1
P’
2
X
3. A polynomial, y = a0 + a1 x + a2 x2 + x3 , passes through the points (−1, 0), (1, 2), (2, 3).
(4 pts) A) Write the augmented matrix of a linear system whose solution is the coefficients a0 , a1 , a2 .
(4 pts) B) Solve the system and write the polynomial that passes through these points.
(over)
Version SAMPLE
Y
4. Consider the data (x,y)= (−1, 3), (0, 0), (2, 2), (3, 2).
(6 pts) A) Find the degree at most 2 least squares
approximation f (x) to the data.
1
(2 pts) B) Sketch f (x) and the data.
1
5. Two plane coordinate systems C and D are related as in
the figure.
(5 pts) A) Write the (homogeneous) transition matrix T(D←C)
from the C- to the D-coordinates.
−1
(next)
1
1
YD
(1 pts) C) Locate P on the figure.
YC
XD
(3 pts) B) Compute the D-coordinates of the point P with Ccoordinates (2, 0).
X
−2
XC
Version SAMPLE
6. Let A be the adjacency matrix of the indicated digraphA
(2 pts) A) Compute A.
(2 pts) B) Let wn be the number of closed walks of length n starting
at vertex 4 in A. Determine the values w1 , w2 ,w3 and w4 .
4
3
1
2
(8 pts) C) Compute the minimal polynomial of A.
Write the recurrence relation for wn .
Use it to determine w5 .


−17 −30 45
7. The matrix A =  25
48 −75  has minimal polynomial µA (x) = (x − 3)(x + 2)
10
20 −32
(6 pts) A) Find its eigenvalues and all associated basic eigenvectors.
(2 pts) B) Write the associated matrix P and diagonal matrix D such that AP = P D.
(over)
Version SAMPLE
8. A Markov chain model has transition matrix
(4 pts) A) Sketch the graph of the characteristic polynomial pA (x) of A
labelling the axes, x-intercept and y-intercept.
Determine the absolute value of each of the eigenvalues of A.


0 0.2 0.8
A =  0.8 0 0.2 .
0.2 0.8 0
(2 pts) B) Determine the steady state distribution of this model.
9. The world wildlife fund is studying the big horned sheep population sampling every 5 years. They observe that big
horned sheep 0-5 years old produce 0.2 offspring on average and 50% survive the sample period. Big horned sheep 5-10
years old produce 1.0 offspring per sample period on average and 100% survive the sample period. Big horned sheep
over 10 years produce an average of 0.6 offspring per sample period and none survive the sample period.
(3 pts) A) Find the transition matrix A of this model of the big horned sheep population.
(2 pts) B) Determine the spectral radius of A.
(2 pts) C) According to this model, what can be said about the big horned sheep population over the long term?
(next)
Version SAMPLE
2
(8 pts) 10. Suppose M is the indicated block matrix product
Z
A
−1
!
A
A
A AT
Z
2A
Express the determinant of M in terms of det(A). Explain your steps!
Y
1 −3
.
11. Let A =
−9 −5
(2 pts) A) Finish the Transform Plot of the linear function fA by sketching the image of of the indicated triangle T .
(5 pts) B) Are ANY of the 3 sides of T eigenvectors?
If so, what are the associated eigenvalues?
1
4
3
1
-3
-1
(over)
1
3 X
!
, where Z is a 2 by 2 matrix of zeros.
Y
(4 pts) 12.
A Complete the transform plot for the matrix A =
2
0
Y
B Do these two grids determine the same lattice? Why.
(6 pts) 13. The formula for y that results from solving the indicated
equation for y is the same as your answer to problem 3 part
b on this test. WITHOUT actually evaluating the determinant, EXPLAIN this fact using properties of determinants.
c
2008
Department of Mathematics Colorado State University
Version SAMPLE
0
.
−1
X

1
 1
0 = det 
 1
1
−1
1
x x2
1
1
2
4
X

1
y − x3 


1
−5