Math 20F Midterm 1 Review Packet
Sample Midterm 1 (Minor 2012)
Math 20F Midterm 1 Review Packet
Sample Midterm 2
Math 20F Midterm 1 Review Packet
Other Midterm Problems
Math 20F Midterm 1 Review Packet
Math 20F Midterm 1 Review Packet
Problems 1 – 5: Prove if the statement is true. If the statement is false, either prove it or give a
counter example. For each statement that is false, what condition would you add to make it true?
1. (A  B)2  A2  2AB  B2
2. If AB  B , then A  I .
3. If A2  0 , then A  0 .
4. If AB  AC and A  0 , then B  C .

 5. If A and B are square such that AB  0 , then A  0 or B  0 .
 6. For (a) and (b) below,
 is A invertible?
2
a) A  0
b) A2  A  I
Review: (Harder) Computational problems
Problems 1 – 2 (ignore the 5x5 matrix on the right hand side of the augmented bar)
Let A  a1  a 7  .
A
I 55 
1 1 1 0
1  2
0
 0
0  2
 0
1 1 0  3
2

 
 0  3  1  9  1  23  3 I 55  ~ 0

 
5 2 16
1
36
5
0
 0
0 12 5 39
 0
1
79 11
1
0
0
0
0
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
1. Express a 4 , a6 , and a 7 as a linear combination of the other column vectors.
2. Does the system Ax  b have a solution for every b? Give a proof or a counterexample.
3. Given the equations below, determine if A is invertible.
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
4. Let A be the following matrix.
0
0
0

0
1
Math 20F Midterm 1 Review Packet
 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 (not invertible), find all solutions to Ax  0 .
5. Let A and A1 be the following matrices. For each B, find B 1 by looking at the relationship
between B and A.
1
0
 2
 1 2 3 


1
A   4  1  3  ,
A   1
4
6 
 3
 1
1
2 
1
2 
 4  1  3 
1
0 
a) B   2
 3
1
2 
 2
b) B   4
 30
 2
c) B   4
 23
1
1
10
1
1
11
0
 3 
20 
0
 3 
2 
6. 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 .
7. Let A be the below matrix.
 1 1 1 1  1
0
1 0 0 0

A
~
2 0 23
0  0

 
 1  1 1  1 0
a) Verify that A is invertible.
1
1
1
1
0
0
0  43  2

0
0  2
Math 20F Midterm 1 Review Packet
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)
b)
c)
d)
e)
f)
g)
Given the system, Ax  b , indicate if each statement is true or false and explain.
If there is a unique solution, the columns of A are independent.
If there is more than 1 solution, then the columns of A are dependent.
If there are no solutions, then the columns of A do not span R m .
If the columns of A are independent, then there is a unique solution.
If the columns of A are dependent, then there are infinitely many solutions.
If the columns of A do not span R m , then there are no solutions.
If the columns of A span R m , then there is a unique solution.
2. Select the right choice and explain.
i) The system must have a nontrivial solution.
ii) The system cannot have a nontrivial solution.
iii) Both choice i) and choice ii) are possible depending on A.
a) Suppose Ax  0 is a system of 3 linear homogeneous equations in 5 variables.
b) Suppose Ax  0 is a system of 5 linear homogeneous equations in 3 variables.
3. 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 the # of pivot columns < 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 the # of pivot columns = m, then there is exactly one solution.
e) If Ax  b is consistent and the # of pivot columns < n, then there are infinitely many
solutions.
f) If the # of pivot columns = n, then Ax  b is consistent for every b.
4. A is an m by n matrix has r pivot columns. What is the relationship between m, n, and r in
each case?
a) A has an inverse.
b) Ax  b has a unique solution for every b in Rm.
c) Ax  b has a unique solution for some, but not all b in Rm.
d) Ax  b has infinitely many solution for every b in Rm.
Review: Proofs
1. Ax  b1 and Ax  b2 are both consistent systems. Is Ax  b1  b2 consistent and why?
2. Let A  span{u, v} and let B  span{u, v, u  v} . Prove A  B .
Math 20F Midterm 1 Review Packet
3. A and B are square matrices. If AB is invertible, prove the following:
a) B is invertible.
b) A is invertible.
4. Suppose S  {v1 , , v n } is a linearly independent in R n and it spans 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 } spans R n .