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MATH1014 Quizzes

Quiz 8: Inverses of matrices
Question 1 Questions
Which of the following is the inverse of the matrix A = 30 0 2 ? Exactly one option must be correct)
a)
1 30 0 1 2
b)
1 20 0 1 3
c)
3 0 0 2
d)
20 0 3
e)
2 0 0 3
f)
02 3 0

Choice (a) is correct!
Choice (b) is incorrect
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Choice (c) is incorrect
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Choice (d) is incorrect
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Choice (e) is incorrect
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Choice (f) is incorrect
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Which of the following is the inverse of the matrix A = 5 7 3 4 ? Exactly one option must be correct)
a)
1 5 1 7 1 3 1 4
b)
5 73 4
c)
4 7 3 5
d)
47 3 5
e)
47 3 5
f)
The inverse of A does not exist.

Choice (a) is incorrect
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Choice (b) is incorrect
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Choice (c) is incorrect
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Choice (d) is incorrect
Check by multiplying this answer by A. The result is not the 2 × 2 identity matrix.
Choice (e) is correct!
Choice (f) is incorrect
The inverse does exist! Try again.
Suppose that A and B are 2 × 2 matrices and that AB = BA = 20 0 2 . Which of the following statements are correct? (More than one option may be correct.) (Zero or more options can be correct)
a)
A1 = B
b)
B1 = A
c)
A1 = 1 2B
d)
B1 = 1 2A
e)
A1 = 2B
f)
B1 = 2A
g)
Neither A1 nor B1 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.
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.
A sequence of elementary row operations transforms the augmented matrix A|I into
130123 0 1 2 1 0 2 001231 .
Find A1. Exactly one option must be correct)
a)
123 1 0 2 231
b)
130 0 1 2 001
c)
102033 6 0 2 3 1
d)
A1 is undefined.
e)
2 2 3 3 6 0 3 0 1

Choice (a) is incorrect
You still need to perform two operations in order to reduce the left hand matrix to the identity matrix.
Choice (b) is incorrect
This is the left hand matrix. It must be reduced to the identity, and then the right hand matrix is the inverse.
Choice (c) is correct!
Choice (d) is incorrect
The inverse is defined. Try again.
Choice (e) is incorrect
The two operations that need to be performed are R2 R2 2R3 and R1 R1 3R2, in that order.
Let A = 312 2 3 4 1 4 5 . The inverse of A is: Exactly one option must be correct)
a)
undefined
b)
B = 312 2 3 4 1 4 5
c)
C = 1 311 2 1 2 1 3 1 4 1 1 4 1 5
d)
D = 1617 4 2 1 118 0
e)
E = 1 3 2 6 13 8 511 7

Choice (a) is incorrect
The inverse exists. Try again.
Choice (b) is incorrect
Multiply A by B. The result is not the 3 × 3 identity matrix.
Choice (c) is incorrect
Multiply A by C. The result is not the 3 × 3 identity matrix.
Choice (d) is incorrect
Multiply A by D. The result is not the 3 × 3 identity matrix.
Choice (e) is correct!
Let A = 100 0 1 0 and B = 10 0 1 00 . Which of the following are correct? (Zero or more options can be correct)
a)
AB = 10 0 1
b)
BA = 100 0 1 0 001
c)
A1 = B
d)
B1 = 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.
Correct!
  1. True
  2. False
  3. False
  4. False
  5. True
  6. True
Let A = ab c d and suppose that A1 = 5 23 1 . Consider the system: ax + by = 2 cx + dy = 7 Which of the following is true ? Exactly one option must be correct)
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.

Choice (a) is correct!
Choice (b) is incorrect
A solution exists. Multiply A1 by 2 7 .
Choice (c) is incorrect
The fact that A1 exists tells you that there is a unique solution.
Choice (d) is incorrect
The solution is unique, but these are incorrect values for x and y.
Choice (e) is incorrect
The solution is found by multiplying A1 by 4 1 .
Find x if Ax = 1 1 1 and A1 = 1 2 5 1 3 6 1 3 7 . Exactly one option must be correct)
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.

Choice (a) is incorrect
x is found by multiplying A1 by 1 1 1 .
Choice (b) is correct!
Choice (c) is incorrect
Try again. ( x is found by multiplying A1 by 1 1 1 .)
Choice (d) is incorrect
Try again. ( x is found by multiplying A1 by 1 1 1 .)
Choice (e) is incorrect
There is a solution, given by x = A1 1 1 1
Suppose that A is an invertible matrix, and consider a system of linear equations Ax = b. Which of the following statements are true? (Zero or more options can be correct)
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.
Correct!
  1. True
  2. False
  3. False
  4. False
  5. True
  6. False
Suppose that A is a square matrix, and that A1 does not exist. Which of the following statements are true? (Zero or more options can be correct)
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.
Correct!
  1. False
  2. False
  3. False
  4. True
  5. True
  6. False Homogeneous equations always have at least one solution.