18.06 Practice Problems for Exam 2

18.06 Practice Problems for Exam 2
Exam will cover Sections 4.1 – 6.2 and Section 8.3 (the part about Markov matrices). Electrical networks (Section 8.2), Fourier series (Section 8.5), and differential
equations (Section 6.3) will not be covered on Exam 2.
Problem
1. 
(a) Do 
Gram-Schmidt



 orthogonalization on the vectors
1
0
0
 −1   1   0 

 
 

 0 ,  −1 ,  1 . (First, get an orthogonal basis, and then an
0
0
−1
orthonormal basis.)
15 6
(b) Find the QR decomposition A = QR of the matrix A =
, and
8 61
−1
−1
−1
compute the inverses Q , R and A .
Problem 2. (Problem 2 from Exam 2, Fall’02) (a) Choose c and the last column
of Q so that you have an orthogonal matrix:


1 −1 −1 ∗
−1 1 −1 ∗

Q = c
−1 −1 −1 ∗
−1 −1 1 ∗
(b) Project b = (1, 1, 1, 1)T onto the first column of Q. Then project b onto the
plane spanned by the first two columns.
(c) Suppose that the last column of the 4 by 4 matrix (where the ∗’s are) was
changed to (1, 1, 1, 1)T . Call this new matrix A. If Gram-Schmidt is applied to
the 4 columns of A, what would be the 4 outputs q1 , q2 , q3 , q4 . (Don’t do a lot of
calculations, please.)
Problem 3. (a) (Problem 2(b) from Exam 2, Spring’07) Find the least squares
b for the system
solution x


 
1 0
1
 0 1 x = 1
−1 0
1
(b) (Problem 17 from Section 4.2, page 228) Write down three equations for the
line b = C + Dt to go through b = 7 at t = −1, b = 7 at t = 1, and b = 21 at t = 2.
b = (C, D) and draw the closest line.
Find the least squares solutions x
1
2
Problem 4. (Problem 1 from Exam 2, Spring’07) 

1 1 2
(a) Compute the determinant of the matrix A = 1 3 1.
4 1 1


1 1 1 2
1 1 3 1

(b) Compute the determinant of the matrix B = 
1 4 1 1.
5 1 1 1
(c) Show that the matrix B from (b) is invertible and calculate the entry (1, 4)
of the inverse matrix B −1 .
Problem 5. (Problem 3 from Exam 2, Fall’09)
(a) Find the area of the triangle on the plane R2 with the vertices (1, 1), (2, 3),
(3, 2).
(b) Calculate the determinant of the 4 by 4 matrix:


1 −1 0
0
−1 1 −1 0 

A=
 0 −1 1 −1
0
0 −1 1
(c) Find the inverse of the matrix A from part (b). Check your answer by
multiplying it with A.
Problem 6. Calculate the determinants:

(a)
a
 l
det 
 k
j
b c
0 0
0 0
i h

d
e 

f 
g

(b)


det 


a
p
o
n
m
b
0
0
0
l
c
0
0
0
k

d e
0 f 

0 g 
.
0 h 
j i
Problem 7. Here is a 4 × 4 matrix filled with the numbers 1, 2, 3, ..., 42 by going
from left to right along the first row, then backwards along the second row, then
forward again along the third, etc.:


1 2 3 4
 8 7 6 5 


 9 10 11 12  .
16 15 14 13
Explain why the determinant of this matrix is 0.
Problem 8. (a) This here is a projection matrix on some subspace of R4 :


3/4 −1/4 1/4
1/4
 −1/4 1/4 −1/4 1/4 


 1/4 −1/4 1/4 −1/4  .
1/4
1/4 −1/4 3/4
3
Find a basis of this subspace.


1
(b) Compute the projection matrix P on the line spanned by the vector  2 
−1
in R3 .
(c) Diagonalize P . (Hint: If you get radicals, you are doing something wrong.)
(d) (Problem 3 from Exam 2, Spring’05) Find the projection of the vector
b = (1, 2, 6)T onto the plane x + y + z = 0 in R3 . (You may want to find a basis of
this 2-dimensional subspace, even an orthogonal basis.)
Problem 9. (Problems 2, 12 from Section 6.1)
(a) Find the eigenvalues and the eigenvectors of these two matrices:
1 4
2 4
A=
and
A+I =
.
2 3
2 4
A + I has the . . . eigenvectors as A. Its eigenvalues are . . . by 1.
(b) Find the three eigenvectors for this matrix P (projection matrices has λ = 1
and 0):


.2 .4 0
Projection matrix
P = .4 .8 0 .
0 0 1
Problem 10. (Problem 2 from Exam 2, Fall’04) Give all values of x for which A
has an eigenvalue equal to 2.


3 2 −1
2
A = 2 x
x −2 3
Problem 11. (Problems 1 and 18 from Section 6.2)
(a) Factor these two matrices into A = SΛS −1 :
1 2
1 1
A=
A=
0 3
3 3
(b) If A = SΛS −1 then A3 = ( )( )( ) and A−1 = ( )( )( ).
(c) Diagonalize A and compute SΛk S −1 to prove this formula for Ak :
1 1 + 3k 1 − 3k
2 −1
k
A=
A =
−1 2
2 1 − 3k 1 + 3k
Problem 12. (a) Find
−9
−10
8
9
20
.


0 1/2 1/3 1/3 1/2
1/4 0 1/3 0
0 



(b) Consider the Markov matrix A = 1/4 1/2 0 1/3 0 
 . Calculate
1/4 0 1/3 0 1/2
1/4 0
0 1/3 0
the limit Ak (0, 1, 0, 0)T as k goes to the infinity.
4
For extra practice try to solve as many of the following problems as you like.
You can also solve other problems from these sections of the textbook and from
past exams.
Problems from textbook:
Section 4.1:
Section 4.2:
Section 4.3:
Section 4.4:
Section 5.1:
Section 5.2:
Section 5.3:
Section 6.1:
Section 6.2:
Section 8.3:
Problems 18 (p. 204), 28 (p. 205).
Problems 11 (p. 215), 19 (p. 216), 20 (p. 216), 33 (p. 217).
Problems 18 (p. 228), 25 (p. 229).
Problems 12 (p. 240), 21 (p. 241), 24 (p. 242).
Problems 2 (p. 251), 13 (p. 252), 15 (p. 253), 24, 27 (p. 254).
Problems 4 (p. 264), 13 (p. 265), 30 (p. 267).
Problems 1, 6 (p. 279), 10 (p. 280), 16 (p. 280).
Problems 6 (p. 293), 9, 10 (p. 294), 15, 21, 22 (p. 295), 30 (p. 296).
Problem 3 (p. 307), 14 (p. 309), 15 (p. 309), 17 (p. 309), 35 (p. 311).
Problem 1 (p. 437), 3 (p. 437), 4 (p. 437), 11 (p. 438).
Problems from past Exams 2:
Past exams are available at http://web.mit.edu/18.06/www/old.shtml
Fall’02, Problem 3. Fall’04, Problem 4. Spring’05, Problems 4, 6. Fall’05,
Problem 1. Spring’07, Problems 2, 3, 4. Fall’09, Problems 1, 2. Fall’10, Problem 1.
Spring’11, Problems 2, 3. Spring’12, Problem 1. Fall’12, Problem 2. Fall’13,
Problems 1, 3. Spring’14, Problems 1, 4.
(Some of these past Exams 2 were given at an earlier date and they did not include eigenvalues and eigenvectors. Our Exam 2 will include eigenvalues/eigenvectors.)