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MATH1002 Quizzes

Quiz 8: Inverses and elementary matrices
Question 1 Questions
Let A = 5 7 3 4 . Which of the following statements are true: (Zero or more options can be correct)
a)
A is singular
b)
A is invertible
c)
A is non-singular
d)
A1 does not exist
e)
The inverse of A is 47 3 5

There is at least one mistake.
For example, choice (a) should be False.
A is singular if A1 does not exist
There is at least one mistake.
For example, choice (b) should be True.
A is invertible is the same as saying that A is non-singular, or that A1 exists. We can see that A is invertible by applying elementary row operations: [AI] = 5 710 3 4 0 1 1 112 3 4 0 1 1 1 1 2 0 1 3 5 .
There is at least one mistake.
For example, choice (c) should be True.
A is non-singular is the same as saying that A is invertible, or that A1 exists.
There is at least one mistake.
For example, choice (d) should be False.
You can see that A is invertible using elementary row operations.
There is at least one mistake.
For example, choice (e) should be True.
You can check that this matrix is the inverse of A by showing that A 47 3 5 = I2 = 47 3 5 A.
Correct!
  1. False A is singular if A1 does not exist
  2. True A is invertible is the same as saying that A is non-singular, or that A1 exists. We can see that A is invertible by applying elementary row operations: [AI] = 5 710 3 4 0 1 1 112 3 4 0 1 1 1 1 2 0 1 3 5 .
  3. True A is non-singular is the same as saying that A is invertible, or that A1 exists.
  4. False You can see that A is invertible using elementary row operations.
  5. True You can check that this matrix is the inverse of A by showing that A 47 3 5 = I2 = 47 3 5 A.
Let A = 312 2 3 4 1 4 5 . The inverse of A (i.e., A1) 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)
none of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is correct!
We could check by carrying out matrix multiplication that none of AB, AC, AD is I, the 3 by 3 identity matrix, so (2), (3) and (4) are all incorrect. (In particular, note that A1 does not consist of the inverses of the entries of A!!) This leaves (1) and (5), and to decide which of them is true, we can augment A by the 3 by 3 identity and reduce: [A|I] = 312100 2 3 4 0 1 0 1 4 5 001 1 4 5 001 2 3 4 0 1 0 312100 1 4 5 00 1 0 5 6 0 2 2 0111310 3 1 4 5 0 0 1 0 1 6502525 011 13 1 0 3 14 5 0 0 1 0165025 25 0015122575 at which point it is clear that there will be a leading 1 in row 3, column 3, so A1 exists. Hence (5) is correct. We do not need to find A1, but continuing the row reduction yields [I|A1], and in fact shows that A1 = 1 3 2 6 13 8 511 7 .
Let A = 1 5 7 2 3 6 11227 . The inverse of A (i.e., A1) is: Exactly one option must be correct)
a)
undefined
b)
B = 1 5 7 2 3 6 11227
c)
C = 1 1 5 1 7 1 2 1 3 1 6 1 1 121 27
d)
D = 1617 4 2 1 118 0
e)
none of the above

Choice (a) is correct!
We could check by carrying out matrix multiplication that none of AB, AC, AD is I, the 3 by 3 identity matrix, so (2), (3) and (4) are all incorrect. (In particular, note that A1 does not consist of the inverses of the entries of A!!) This leaves (1) and (5), and to decide which of them is true, we can augment A by the 3 by 3 identity and reduce: [A|I] = 1 5 7 100 2 3 6 0 1 0 11227001 1 5 7 1 00 0 7 20 2 1 0 0 7 20101 1 5 7 1 00 0 7 20 2 1 0 0 0 0 311 at which point it is clear that there can be no leading 1 in row 3, column 3, so (1) is correct.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Which of the following matrices are elementary matrices? (Zero or more options can be correct)
a)
100 0 0 1 010
b)
100 0 2 1 001
c)
010 0 0 1 100
d)
120 0 1 0 001

There is at least one mistake.
For example, choice (a) should be True.
This matrix corresponds to the elementary row operation R2 R3.
There is at least one mistake.
For example, choice (b) should be False.
This matrix is the product of two elementary matrices corresponding to R2 := 2R2 then R2 := R2 + R3
There is at least one mistake.
For example, choice (c) should be False.
This matrix is product of two elementary matrices corresponding to R1 R3 then R2 R1
There is at least one mistake.
For example, choice (d) should be True.
This matrix to the elementary row operation R1 := R1 + 2R2.
Correct!
  1. True This matrix corresponds to the elementary row operation R2 R3.
  2. False This matrix is the product of two elementary matrices corresponding to R2 := 2R2 then R2 := R2 + R3
  3. False This matrix is product of two elementary matrices corresponding to R1 R3 then R2 R1
  4. True This matrix to the elementary row operation R1 := R1 + 2R2.
The elementary matrix corresponding to the Elementary Row Operation R3 = R3 5R2 on a matrix with 3 rows is: Exactly one option must be correct)
a)
10 0 0 1 5 00 1
b)
100 0 1 5 001
c)
1 0 0 0 1 0 051
d)
100 0 1 0 051
e)
none of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The correct answer is (3), since that is the result of applying the give elementary row operation to the 3 by 3 identity matrix.
Note that 1 0 0 0 1 0 051 ab c d e f = a b c d e 5cf 5d is the matrix we would get by applying the given elementary row operation to ab c d e f .
Choice (d) is incorrect
Choice (e) is incorrect
The elementary matrix corresponding to the Elementary Row Operation R3 = 1 9R3 on a matrix with 4 rows is: Exactly one option must be correct)
a)
100 0 1 0 009
b)
100 0 1 0 001 9
c)
1000 0 1 0 0 0090 0 0 0 1
d)
1000 0 1 0 0 001 90 0001
e)
none of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The correct answer is (4), since that is the result of applying the give elementary row operation to the 4 by 4 identity matrix.
Note that 1000 0 1 0 0 001 90 0 0 0 1 ab c d e f g h = a b c d 1 9e1 9f g h is the matrix we would get by applying the given elementary row operation to ab c d e f g h .
Choice (e) is incorrect
Consider the six elementary matrices below:
E1 = 1000 0 1 0 0 0010 0 2 0 1 ,E2 = 0100 1 0 0 0 0010 0 0 0 1 ,E3 = 1 0 00 0 1 0 0 0 0 10 0 2 0 1 , E4 = 0010 0 1 0 0 1000 0 0 0 1 ,E5 = 1200 0 1 0 0 0 0 10 0 0 0 1 ,E6 = 1200 0 1 0 0 0010 0 0 0 1 .
Which of the following statements are correct? (Zero or more options can be correct)
a)
The inverse of E1 is E2.
b)
The inverse of E1 is E3.
c)
The inverse of E2 is E2.
d)
The inverse of E5 is E6.
e)
The inverse of E4 is E4.
f)
The inverse of E3 is E6.

There is at least one mistake.
For example, choice (a) should be False.
E1 corresponds to the row operation R4 = R4 + 2R2, whereas E2 corresponds to R1R2.
There is at least one mistake.
For example, choice (b) should be True.
E1 corresponds to the row operation R4 = R4 + 2R2 and E3 corresponds to the row operation R4 = R4 2R2.
There is at least one mistake.
For example, choice (c) should be True.
E2 corresponds to R1R2.
There is at least one mistake.
For example, choice (d) should be True.
E5 corresponds to the row operation R1 = R1 2R2, whereas E6 corresponds to R1 = R1 + 2R2.
There is at least one mistake.
For example, choice (e) should be True.
E4 corresponds to R1R3.
There is at least one mistake.
For example, choice (f) should be False.
E3 corresponds to the row operation R4 = R4 2R2, whereas E6 corresponds to R1 = R1 + 2R2.
Correct!
  1. False E1 corresponds to the row operation R4 = R4 + 2R2, whereas E2 corresponds to R1R2.
  2. True E1 corresponds to the row operation R4 = R4 + 2R2 and E3 corresponds to the row operation R4 = R4 2R2.
  3. True E2 corresponds to R1R2.
  4. True E5 corresponds to the row operation R1 = R1 2R2, whereas E6 corresponds to R1 = R1 + 2R2.
  5. True E4 corresponds to R1R3.
  6. False E3 corresponds to the row operation R4 = R4 2R2, whereas E6 corresponds to R1 = R1 + 2R2.
Let A = 311 1 2 3 0 1 1 , then A1 = 1 2 5 1 3 8 1 3 7 .
What sequence of elementary row operations is needed to transform A|I to I|A1 ? Exactly one option must be correct)
a)
R2 := R2 1 3R1, R2 := 3R2, R2 R3, R2 := R2 R3,
R3 := R3 7R2, R1 := R1 + R2, R1 := R1 R3, R1 := 1 3R1
b)
R1 := 1 3R1, R2 := R2 R1, R2 := 3 7R2, R3 := R3 R2
R3 := 7R3, R2 := R2 8 7R3, R1 := R1 + 1 3R2, R1 := R1 1 3R3
c)
R2 := R2 1 3R1, R3 := R3 R2, R2 := 3 7R2, R3 := 7R3,
R2 := R1 R3, R1 := R1 + R2, R1 := R1 R3
d)
All of the above sequences

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
The system of equations 3x y + z = 1, x + 2y + 3z = 1, y + z = 1, can be written in the form Ax = b where
A = 311 1 2 3 0 1 1 ,x = x y z andb = 1 1 1 .
Using A1 (given in question 8) to solve the system of equations, which of the following statements is correct ? Exactly one option must be correct)
a)
y = 10
b)
z = 5
c)
x = 6
d)
None of the above

Choice (a) is correct!
A1b = 610 9 x = 6, y = 10, z = 9.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Let A = abc d e f g hi , and suppose A1 = 13 2 0 3 5 168 . Consider the system ax + by + cz = 5 dx + ey + fz = 1 gx + hy + iz = 4 Which of the following is true ? Exactly one option must be correct)
a)
the system has no solutions;
b)
the system has many solutions;
c)
the system has the unique solution x = 16,y = 23,z = 21;
d)
the system has the unique solution x = 14,y = 22,z = 13;
e)
none of the above.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Since A is invertible, there is the unique solution x y z = A1 5 1 4 = 13 2 0 3 5 168 5 1 4 = 16 23 21 . Hence the correct answer is (3).
Choice (d) is incorrect
Choice (e) is incorrect