WORKSHEET 2 - solutions
Math 53, GSI: Toni Antunovi
c
c
Problem 1. Consider the set of m n matrices whose entry at the intersection
of the rst row and rst column is a. For what values of a is this set a vector
subspace (of the vector space formed by m n m
Quiz 2
Math 54, Section 203
GSI: Toni Antunovi
c
c
February 16, 2011
Solutions of all problems must by accompanied by relevant explanations.
Problem 1. Find the determinant of the matrix
1 2 1 3
3
0
1 0
.
2 1
0 4
0
3
21
Is this matrix invertible?
Solutio
Quiz 3
Math 54, Section 203
GSI: Toni Antunovi
c
c
March 16, 2011
Solutions of all problems must by accompanied by relevant explanations.
Problem 1. Find all eigenvalues of the matrix
4
2 2
1 4
A= 2
1 1/2
1
Find the dimensions of the corresponding eigens
Quiz 4
Math 54, Section 203
GSI: Toni Antunovi
c
c
May 9, 2011
Solutions of all problems must by accompanied by relevant explanations.
Problem 1. Find the general solution to the equation
y 4y + 3 = 0.
Solution:
We write the equation as y 4y = 3. A partic
Quiz 5
Math 54, Section 203
GSI: Toni Antunovi
c
c
April 29, 2011
Solutions of all problems must by accompanied by relevant explanations.
Problem 1. Find the solution to the following initial problem
y + 3y + 3y + y = 0; y (0) = 0, y (0) = 0, y (0) = 2.
S
Quiz 1
Math 54, Section 208
GSI: Toni Antunovi
c
c
Problem 1. Find all solutions to the system
3x1
x1
+ 2x 2
+
x2
x2
7x3
2x3
+
x3
Solution:
2x4
x4
=
=
=
2
4
4
3 2 7 2 2
0 4 . Doing row
The augmented matrix of the system is 1 1 2
01
1 1 4
transformations
Quiz 2
Math 54, Section 208
GSI: Toni Antunovi
c
c
February 16, 2011
Solutions of all problems must by accompanied by relevant explanations.
Problem 1. For each of the following two matrices determine if it is invertible
and if yes nd its inverse.
1
1
a)
Quiz 3
Math 54, Section 208
GSI: Toni Antunovi
c
c
March 16, 2011
Solutions of all problems must by accompanied by relevant explanations.
Problem 1. Find all eigenvalues of the matrix
6 2
2
3
A = 6 1
6
3 1
Find the dimensions of the corresponding eigenspa
Quiz 4
Math 54, Section 208
GSI: Toni Antunovi
c
c
April 13, 2011
Solutions of all problems must by accompanied by relevant explanations.
Problem 1. Find the solution to the dierential equation
y 6y + 9y = 0; y (0) = 0, y (0) = 4.
Solution:
The auxiliary
Quiz 5
Math 54, Section 208
GSI: Toni Antunovi
c
c
April 29, 2011
Solutions of all problems must by accompanied by relevant explanations.
Problem 1. Find the largest interval on which you can conclude that the initial problem has
the unique solution.
1
,
Problem 1. If
A=
2
1
0
, B = 1
1
34
05
1
0 , C =
2
2
3
1
0
determine which of these expressions are well-dened and calculate those which
are.
a) A 2B
b) AB + 2C
c) A1 + BC T
d) C 1 + C T .
Solution:
a) A 2B is not well dened since A is a 2 3 matrix and B
Quiz 1
Math 54, Section 203
GSI: Toni Antunovi
c
c
Problem 1. Find all solutions to the system
3x1
x1
+ 2x 2
+
x2
x3
+ x3
+ x3
Solution:
+
x4
5 x4
2 x4
= 10
= 11
=4
001
1 10
Augmented matrix of this system is 3 2 1 5 11 . By applying
1 1 1 2 4
row trans