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Quiz 8: Inverses of matrices

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Which of the following is the inverse of the matrix A = [3 0] 0 2?
a) [ ] 13 0 0 12   b) [ ] 12 0 0 13
c) [ ] - 3 0 0 - 2   d) [2 0] 0 3
e) [ ] - 2 0 0 - 3   f) [ ] 0 2 3 0

 

Your answer is correct.
Not correct. Choice (b) is false.
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Not correct. Choice (c) is false.
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Not correct. Choice (d) is false.
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Not correct. Choice (e) is false.
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Not correct. Choice (f) is false.
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
 

Question 2

 
 
Which of the following is the inverse of the matrix A = [ ] - 5 7 3 - 4?
a) [ ] -51 17 13 -14   b) [ ] 5 - 7 - 3 4
c) [ ] - 4 7 3 - 5   d) [ ] - 4 - 7 - 3 - 5
e) [ ] 4 7 3 5   f) The inverse of A does not exist.

 

Not correct. Choice (a) is false.
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Not correct. Choice (b) is false.
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Not correct. Choice (c) is false.
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Not correct. Choice (d) is false.
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Your answer is correct.
Not correct. Choice (f) is false.
The inverse does exist! Try again.
 

Question 3

 
 
Suppose that A and B are 2 × 2 matrices and that AB = BA = [2 0] 0 2. Which of the following statements are correct? (More than one option may be correct.)
a) A-1 = B   b) B-1 = A
c) A-1 = 12B   d) B-1 = 12A
e) A-1 = 2B   f) B-1 = 2A
g) Neither A-1 nor B-1 exist.

 

There is at least one mistake.
For example, choice (a) should be false.
Note that AB = 2I2.
There is at least one mistake.
For example, choice (b) should be false.
Note that AB = 2I2.
There is at least one mistake.
For example, choice (c) should be true.
There is at least one mistake.
For example, choice (d) should be true.
There is at least one mistake.
For example, choice (e) should be false.
Note that (2B)A = 4I2.
There is at least one mistake.
For example, choice (f) should be false.
Note that (2A)B = 4I2.
There is at least one mistake.
For example, choice (g) should be false.
Both inverses exist. Try again.
Your answers are correct
  1. False. Note that AB = 2I2.
  2. False. Note that AB = 2I2.
  3. True.
  4. True.
  5. False. Note that (2B)A = 4I2.
  6. False. Note that (2A)B = 4I2.
  7. False. Both inverses exist. Try again.
 

Question 4

 
 
A sequence of elementary row operations transforms the augmented matrix [A |I] into
⌊ 1 3 0|1 2 3 ⌋ ⌈ 0 1 2|1 0 2 ⌉ 0 0 1|2 3 1.
Find A-1.
a) ⌊ ⌋ 1 2 3 ⌈1 0 2⌉ 2 3 1   b) ⌊ ⌋ 1 3 0 ||0 1 2|| ⌈0 0 1⌉
c) ⌊ ⌋ ⌈10 20 3⌉ - 3 - 6 0 2 3 1   d) A-1 is undefined.
e) ⌊- 2 2 - 3⌋ ⌈- 3 - 6 0 ⌉ - 3 0 1

 

Not correct. Choice (a) is false.
You still need to perform two operations in order to reduce the left hand matrix to the identity matrix.
Not correct. Choice (b) is false.
This is the left hand matrix. It must be reduced to the identity, and then the right hand matrix is the inverse.
Your answer is correct.
Not correct. Choice (d) is false.
The inverse is defined. Try again.
Not correct. Choice (e) is false.
The two operations that need to be performed are R2 → R2 - 2R3 and R1 → R1 - 3R2 , in that order.
 

Question 5

 
 
Let A = ⌊ ⌋ - 3 - 1 - 2 ⌈ 2 3 4⌉ 1 4 5. The inverse of A is:
a) undefined   b) B = ⌊- 3 - 1 - 2⌋ ⌈ 2 3 4 ⌉ 1 4 5
c) C = ⌊- 13 1 - 12⌋ | | || 12 13 14|| |⌈ |⌉ 1 14 15   d) D = ⌊ ⌋ 1 - 6 17 ⌈ 4 - 2 1⌉ 11 - 8 0
e) E = ⌊ 1 3 - 2⌋ ⌈ 6 13 - 8⌉ - 5 - 11 7

 

Not correct. Choice (a) is false.
The inverse exists. Try again.
Not correct. Choice (b) is false.
Multiply A by B. The result is not the 3 × 3 identity matrix.
Not correct. Choice (c) is false.
Multiply A by C. The result is not the 3 × 3 identity matrix.
Not correct. Choice (d) is false.
Multiply A by D. The result is not the 3 × 3 identity matrix.
Your answer is correct.
 

Question 6

 
 
Let A = [ ] 1 0 0 0 1 0 and B = ⌊ 1 0⌋ ⌈ 0 1⌉ 0 0. Which of the following are correct?
a) AB = [ ] 1 0 0 1   b) BA = ⌊1 0 0⌋ ⌈0 1 0⌉ 0 0 1
c) A-1 = B   d) B-1 = A
e) A is not invertible.   f) B is not invertible.

 

There is at least one mistake.
For example, choice (a) should be true.
There is at least one mistake.
For example, choice (b) should be false.
There is at least one mistake.
For example, choice (c) should be false.
There is at least one mistake.
For example, choice (d) should be false.
There is at least one mistake.
For example, choice (e) should be true.
There is at least one mistake.
For example, choice (f) should be true.
Your answers are correct
  1. True.
  2. False.
  3. False.
  4. False.
  5. True.
  6. True.
 

Question 7

 
 
Let A = [ ] a b c d and suppose that A-1 = [ ] 5 - 2 - 3 1. Consider the system:
ax + by = 2
cx + dy = 7
Which of the following is true ?
a) The system has the unique solution x = -4, y = 1.   b) The system has no solution.
c) The system has infinitely many solutions.   d) The system has the unique solution x = 24, y = 13.
e) It is not possible to say anything about the solution unless we know the values of a, b, c and d.

 

Your answer is correct.
Not correct. Choice (b) is false.
A solution exists. Multiply A-1 by [ ] - 4 1.
Not correct. Choice (c) is false.
The fact that A-1 exists tells you that there is a unique solution.
Not correct. Choice (d) is false.
The solution is unique, but these are incorrect values for x and y.
Not correct. Choice (e) is false.
The solution is found by ultiplying A-1 by [- 4] 1.
 

Question 8

 
 
Find x if Ax = ⌊ 1 ⌋ ⌈ 1 ⌉ - 1 and A-1 = ⌊ 1 - 2 5⌋ ⌈ 1 - 3 6⌉ - 1 3 - 7.
a) It is not possible to find x without knowing what A is.   b) x = ⌊- 6⌋ ⌈- 8⌉ 9
c) x = ⌊ 6⌋ ⌈- 8⌉ - 5   d) x = ⌊ ⌋ - 6 ⌈- 8⌉ 11
e) The system Ax = ⌊ 1 ⌋ ⌈ 1 ⌉ - 1 is inconsistent. There are no solutions for x.

 

Not correct. Choice (a) is false.
x is found by multiplying A-1 by ⌊ 1 ⌋ ⌈ 1 ⌉ - 1.
Your answer is correct.
Not correct. Choice (c) is false.
Try again. ( x is found by multiplying A-1 by ⌊ 1⌋ ⌈ 1⌉ - 1.)
Not correct. Choice (d) is false.
Try again. ( x is found by multiplying A-1 by ⌊ ⌋ 1 ⌈ 1⌉ - 1.)
Not correct. Choice (e) is false.
There is a solution, given by x = A-1⌊ 1 ⌋ ⌈ 1 ⌉ - 1
 

Question 9

 
 
Suppose that A is an invertible matrix, and consider a system of linear equations Ax = b. Which of the following statements are true?
a) The system has a unique solution.   b) The system has infinitely many solutions.
c) The system has no solutions.   d) It depends on A and b. Any of (a), (b) or (c) could be true.
e) If b = 0 the only solution is the trivial solution.   f) If b = 0 the system has infinitely many solutions.

 

There is at least one mistake.
For example, choice (a) should be true.
There is at least one mistake.
For example, choice (b) should be false.
There is at least one mistake.
For example, choice (c) should be false.
There is at least one mistake.
For example, choice (d) should be false.
There is at least one mistake.
For example, choice (e) should be true.
There is at least one mistake.
For example, choice (f) should be false.
Your answers are correct
  1. True.
  2. False.
  3. False.
  4. False.
  5. True.
  6. False.
 

Question 10

 
 
Suppose that A is a square matrix, and that A-1 does not exist. Which of the following statements are true?
a) Any system of linear equations Ax = b will have a unique solution.   b) Any system of linear equations Ax = b will have infinitely many solutions.
c) Any system of linear equations Ax = b will be inconsistent.   d) A system of linear equations Ax = b could have infinitely many solutions, or no solution, depending on b.
e) A system of linear equations Ax = 0 will have infinitely many solutions.   f) A system of linear equations Ax = 0 will have no solution.

 

There is at least one mistake.
For example, choice (a) should be false.
There is at least one mistake.
For example, choice (b) should be false.
There is at least one mistake.
For example, choice (c) should be false.
There is at least one mistake.
For example, choice (d) should be true.
There is at least one mistake.
For example, choice (e) should be true.
There is at least one mistake.
For example, choice (f) should be false.
Homogeneous equations always have at least one solution.
Your answers are correct
  1. False.
  2. False.
  3. False.
  4. True.
  5. True.
  6. False. Homogeneous equations always have at least one solution.