Sample Midterm 1 (Minor 2012)
Sample Midterm 2 (2010)
Sample Midterm 3
Other Midterm Problems
Review: Computational problems
Problems 1 – 5
Let A  a1  a 7  .
A
I 55 
1 1 1 0
1  2
0
 0 1
0  2
 0 0
1 1 0  3
2

 
 0  3  1  9  1  23  3 I 55  ~ 0 0

 
5 2 16
1
36
5
0
 0 0
0 12 5 39
 0 0
1
79 11
1. Find the dimension of each of the following spaces.
a) Row ( A)
b) Col ( A)
c) Nul ( A)
0
1
0
0
0
2
3
0
0
0
0
0
1
0
0
4
5
6
0
0
0 1 1 0 0
2  2 1 0 0
1
5 4 1 0
0
4 3
1 1
0 17 13
1 0
2. Verify the following relationship numerically.
a) rk ( A)  dim[ Nul ( A)]  n
3.
a)
b)
c)
Find a basis for each of the following spaces.
Row ( A)
Col ( A)
Nul ( A)
4. Express a 4 , a6 , and a 7 as a linear combination of the other column vectors.
5. Does the system Ax  b have a solution for every b? Give a proof or a counterexample.
6. How can you tell that det( A)  0 quickly?
 1 2 3 4
  5 6 7 8

a) A  
 9 10 11 12


 0 0 0 0
 1 2 0 3
  6 5 0 4

b) A  
  7 8 0 9


 12 11 0 10
 1 2 3 4
 5 6 7 8

c) A  
 9 10 11 12


10 20 30 40
0
0
0

0
1
 1 2 3
7. Given A  4 6 7 and det( A)  2 , find det( B ) .
 5 8 9
 1 2 41 3
 1
0 0 42 0
d) B   4

a) B  
4 6 43 7 
15


 5 8 99 9
1
 5 8 9
e) B  2
b) B  4 6 7
 3
 1 2 3
f) BA  I
10
 1 2 3
g) B  40
c) B   4 6 7
50
50 80 90
3
6 7
28 39
4 5
6 8
7 9
2
2 3 2 1 3
6 7 6 4 7
8 9 8 5 9
8. Given the equations below, find det( A) .
0   8 
1  1
0   4








a) A4   0 , A3  0 , A0   0 
4 10
5  2
2 5
1  1
0   4
0 3








b) A4   0 , A3  0 , A0  5
5  2
2 5
4 3
9. Let A be the following matrix.
 0 2 1 0
 0 1 0 0

A
  3 2 3


 1 5 1 1
a) For what values of  is A invertible?
b) Assuming A is singular, find the rk ( A) , dim[ Nul ( A)] , and the Nul ( A) .
10. Let A be a 4  4 matrix of all ones.
1 1 1 1
1 1 1 1

A
1 1 1 1


1 1 1 1
a) Show A2  4 A .
b) Let B  A  2 I . Show 8B  B 2  12 I  0
c) Find B 1 .
11. Let A be the below matrix.
1
1
 1 1 1 1  1 1
0


1 0 0  0 1
0
0

A
~
2 0 23
0 0 0  43  2

 

0  2
 1  1 1  1 0 0
a) Verify that A is invertible.
b) Suppose b1, b2, b3, and b4 are real numbers. Show that there is exactly one P3 polynomial
such that the following equations are true.
p(1)  b1 ,
p(0)  b2 ,
1

1
p( x)dx  b3 , and p(1)  b4
Review: Conceptual questions
1. A is an m by n matrix. Justify your answer with a proof if the statement is true, or give a
counter-example or proof if the statement is false.
a) If Ax  b is not consistent, then rk ( A)  m .
b) If Ax  b is consistent and n  m , then there are infinitely many solutions.
c) If Ax  b is consistent and n  m , then there is exactly one solution.
d) If Ax  b is consistent and rk ( A)  m , then there is exactly one solution.
e) If Ax  b is consistent and rk ( A)  n , then there are infinitely many solutions.
f) If rk ( A)  n , then Ax  b is consistent for every b.
2.
a)
b)
c)
d)
A is an m by n matrix of rank r. What is the relationship between m, n, and r in each case?
A has an inverse.
Ax  b has a unique solution for every b in Rm.
Ax  b has a unique solution for some, but not all b in Rm.
Ax  b has infinitely many solution for every b in Rm.
Review: Proofs
1. Suppose S  {v1 , , v n } is a basis of R n , and A is an n  n invertible matrix.
a) Prove the following set B  { Av1 , , Av k } is independent when k  n .
b) Prove the following set C  { Av1 ,, Av n } is a basis for R n .
2. Let A  span{u, v} and let B  span{u, v, u  v} . Prove A  B .
3. Suppose A is m n . If Col ( A)  Row( A) , prove m  n .
4. Suppose A 2  0 . Prove A is not invertible.