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Quiz 5: Row echelon form

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Question 1

 
 
Which of the following arrays are 3 × 2 matrices? (More than one answer may be correct.)
a) [ ] 3 - 2 - 1 10 0 22   b) ⌊ ⌋ 3 10 ⌈- 2 0⌉ - 1 22
c) ⌊ ⌋ ⌈2 5 ⌉ 3 - 9 5   d) [3 - 2 - 1] 10 - 3
e) ⌊3 - 2 - 1⌋ ⌈3 - 2 - 1⌉ 4 10 - 3

 

There is at least one mistake.
For example, choice (a) should be false.
This is a 2 × 3 matrix (2 rows, 3 columns).
There is at least one mistake.
For example, choice (b) should be true.
This matrix has 3 rows and 2 columns.
There is at least one mistake.
For example, choice (c) should be false.
This is not a matrix (i.e., is not a rectangular array) since not all rows have the same numbers of entries (and not all columns have the same number of entries).
There is at least one mistake.
For example, choice (d) should be false.
This is not a matrix (i.e., is not a rectangular array) since not all rows have the same numbers of entries (and not all columns have the same number of entries).
There is at least one mistake.
For example, choice (e) should be false.
This is a 3 × 3 matrix.
Your answers are correct
  1. False. This is a 2 × 3 matrix (2 rows, 3 columns).
  2. True. This matrix has 3 rows and 2 columns.
  3. False. This is not a matrix (i.e., is not a rectangular array) since not all rows have the same numbers of entries (and not all columns have the same number of entries).
  4. False. This is not a matrix (i.e., is not a rectangular array) since not all rows have the same numbers of entries (and not all columns have the same number of entries).
  5. False. This is a 3 × 3 matrix.
 

Question 2

 
 
Which of the following is the elementary row operation which transforms the matrix
⌊ 2 4 2 - 1⌋ ⌊2 4 2 - 1⌋ A = ⌈ 2 5 1 6 ⌉ into B = ⌈0 1 - 1 7 ⌉ ? 1 2 0 1 1 2 0 1
a) R2 := R1 - 2R2   b) R1 := R1 - R2
c) R1 := R2 - R1   d) R1←→R3
e) R2 := R2 - R1

 

Not correct. Choice (a) is false.
This is not an elementary row operation.
Not correct. Choice (b) is false.
This will alter row 1 and leave row 2 unchanged. So applying this elementary row operation to A will give the matrix
⌊ ⌋ ⌈0 - 1 - 1 - 7⌉ 2 5 1 6 1 2 0 1
give B from A.
Not correct. Choice (c) is false.
This is not an elementary row operation.
Not correct. Choice (d) is false.
This operation interchanges row 1 and row 3. Applying this operation to A gives the matrix
⌊1 2 0 1 ⌋ ⌈2 5 1 6 ⌉ 2 4 2 - 1
Your answer is correct.
 

Question 3

 
 
Which of the following matrices is the augmented matrix for the following system of equations:
w + 2x - 5 = 0
2w + 3x + 6z + 1 = 0
3x - 7 - 2z = 0
in the four unknowns w,x,y and z?
a) ⌊ ⌋ 1 2 0 5 ⌈ 2 3 6 - 1 ⌉ 0 3 - 2 7   b) ⌊ | ⌋ ⌈ 0 1 2 |- 5 ⌉ 2 3 6 | 1 0 3 - 7 - 2
c) ⌊ | ⌋ ⌈ 1 2 0 0 | 5 ⌉ 2 3 0 6 |- 1 0 3 0 2 7   d) ⌊ 1 2 0|5 ⌋ ⌈ 2 3 6|1 ⌉ 0 3 2|7
e) ⌊ 1 2 0 0 | 5 ⌋ ⌈ 2 3 0 6 |- 1 ⌉ 0 3 0 - 2 | 7

 

Not correct. Choice (a) is false.
The augmented matrix should have 5 columns — one for each of the 4 unknowns and one for the constants on the right hand side.
Not correct. Choice (b) is false.
This is just the array of coefficients in the equations.
Not correct. Choice (c) is false.
One of the coefficients in the fourth column is wrong.
Not correct. Choice (d) is false.
Even though the variable y does not appear in any of the equations it still must have a column in the augmented matrix.
Your answer is correct.
The system of equations can be rewritten with the constant terms on the right hand side, giving
w + 2x = 5
2w + 3x + 6z = -1
3x - 2z = 7
 

Question 4

 
 
Which of the following matrices are in row echelon form? (More than one answer may be correct.)
a) ⌊ ⌋ 1 3 0 5 ||0 0 1 0|| ⌈0 0 0 2⌉ 0 0 0 0   b) ⌊1 3 0 5⌋ |0 0 1 0| |⌈0 0 0 1|⌉ 0 0 0 0
c) ⌊0 1 1 2⌋ |0 1 1 0| |⌈0 0 0 0|⌉ 0 0 0 0   d) ⌊ ⌋ 1 3 6 5 ||0 0 1 0|| ⌈0 0 0 0⌉ 0 0 0 0
e) ⌊ ⌋ |1 1 1 2| |0 0 0 0| ⌈0 1 1 0⌉ 0 0 0 0

 

There is at least one mistake.
For example, choice (a) should be false.
The first non-zero in row 3 is not 1, so this is in row echelon form.
There is at least one mistake.
For example, choice (b) should be true.
This is in row echelon form because the first non–zero entry in each non–zero row is equal to 1, and each leading 1 is in a later column of the matrix that the leadings 1s in previous rows, with the zero rows occurring last.
There is at least one mistake.
For example, choice (c) should be false.
The leading 1s in rows 1 and 2 appear in the same column.
There is at least one mistake.
For example, choice (d) should be true.
This is in row echelon form. Note, however, that this matrix is not in reduced row echelon form since the entry in row 1, column 3 is non–zero.
There is at least one mistake.
For example, choice (e) should be false.
The zero rows occur at the bottom of matrices which are in row echelon form.
Your answers are correct
  1. False. The first non-zero in row 3 is not 1, so this is in row echelon form.
  2. True. This is in row echelon form because the first non–zero entry in each non–zero row is equal to 1, and each leading 1 is in a later column of the matrix that the leadings 1s in previous rows, with the zero rows occurring last.
  3. False. The leading 1s in rows 1 and 2 appear in the same column.
  4. True. This is in row echelon form. Note, however, that this matrix is not in reduced row echelon form since the entry in row 1, column 3 is non–zero.
  5. False. The zero rows occur at the bottom of matrices which are in row echelon form.
 

Question 5

 
 
Which of the following matrices are in row echelon form? (More than one answer may be correct.)
a) ⌊ ⌋ 1 4 0 5 ||0 0 1 0|| ⌈0 0 0 0⌉ 0 0 0 1   b) ⌊ ⌋ 1 4 0 5 ||1 0 1 0|| ⌈0 0 0 1⌉ 0 0 0 0
c) ⌊1 4 0 5⌋ |0 0 - 1 0| |⌈0 0 0 0|⌉ 0 0 0 0   d) ⌊ ⌋ 1 2 - 1 6 ||0 1 - 1 4|| ⌈0 0 4 1⌉ 0 0 0 7
e) ⌊1 2 - 7 6⌋ |0 1 - 1 4| |⌈0 0 1 4|⌉ 0 0 0 1

 

There is at least one mistake.
For example, choice (a) should be false.
The (only) zero row is not the last row, so A is not in row echelon form.
There is at least one mistake.
For example, choice (b) should be false.
The leading 1 in row 2 is in the same column as the leading 1 is in row 1, so this matrix is not in row echelon form.
There is at least one mistake.
For example, choice (c) should be false.
The first non-zero in row 2 is not 1, so this matrix is not a row echelon matrix.
There is at least one mistake.
For example, choice (d) should be false.
The leading entries in rows 3 and 4 are not equal to 1 so this matrix is not in row echelon form.
There is at least one mistake.
For example, choice (e) should be true.
Any “triangular” matrix with 1s down its diagonal is automatically in row echelon form.
Your answers are correct
  1. False. The (only) zero row is not the last row, so A is not in row echelon form.
  2. False. The leading 1 in row 2 is in the same column as the leading 1 is in row 1, so this matrix is not in row echelon form.
  3. False. The first non-zero in row 2 is not 1, so this matrix is not a row echelon matrix.
  4. False. The leading entries in rows 3 and 4 are not equal to 1 so this matrix is not in row echelon form.
  5. True. Any “triangular” matrix with 1s down its diagonal is automatically in row echelon form.
 

Question 6

 
 
Which of the statements below correctly describes the solutions of the system of equations with the following augmented matrix?
⌊ 1 1 - 3 0 |0 ⌋ | 0 0 1 5 |1 | |⌈ 0 0 1 6 |2 |⌉ 0 0 0 1 |2
a) The solution is unique.
b) There are infinitely many solutions.
c) The system is inconsistent, there are no solutions.
d) There is not enough information to decide.

 

Not correct. Choice (a) is false.
Hint: apply suitable elementary row operations to the augmented matrix.
Not correct. Choice (b) is false.
Hint: apply suitable elementary row operations to the augmented matrix.
Your answer is correct.
Reduce the matrix to row echelon form.
⌊ 1 1 - 3 0 |0 ⌋ | 0 0 1 5 |1 | |⌈ 0 0 1 6 |2 |⌉ 0 0 0 1 |2
R3:=R3-R2 --------→⌊ 1 1 - 3 0 |0 ⌋ | 0 0 1 5 |1 | |⌈ 0 0 0 1 |1 |⌉ 0 0 0 1 |2
   
R4:=R4-R3 --------→⌊ 1 1 - 3 0 |0 ⌋ | 0 0 1 5 |1 | |⌈ 0 0 0 1 |1 |⌉ 0 0 0 0 |1
 
The matrix is in reduced row echelon form and there is a leading one in the last column so the system is inconsistent and has no solution.
Not correct. Choice (d) is false.
It is always possible to decide such questions by reducing the augmented matrix to row echelon form.
 

Question 7

 
 
Which of the statements below correctly describes the solutions of the system of equations with the following augmented matrix?
⌊ | ⌋ 1 2 7 1 |3 || 0 1 - 5 7 |1 || ⌈ 0 1 - 4 10 |2 ⌉ 0 0 1 4 2
a) The solution is unique.
b) There are infinitely many solutions.
c) The system is inconsistent, there are no solutions.
d) There is not enough information to decide.

 

Your answer is correct.
Reduce the matrix to row echelon form.
⌊ | ⌋ 1 2 7 1 |3 || 0 1 - 5 7 |1|| ⌈ 0 1 - 4 10 |2⌉ 0 0 1 4 2
-R-3-:=-R-3---R-2→⌊ | ⌋ 1 2 7 1 |3 || 0 1 - 5 7 |1 || ⌈ 0 0 1 3 |1 ⌉ 0 0 1 4 2
   
-R-4-:=-R-4---R-3→⌊ | ⌋ 1 2 7 1 |3 || 0 1 - 5 7 |1 || ⌈ 0 0 1 3 |1 ⌉ 0 0 0 1 1
 
The matrix is consistent and there is a leading one in each column, except the last, so the system has a unique solution.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
It is always possible to decide such questions by reducing the augmented matrix to row echelon form.
 

Question 8

 
 
Which of the following correctly describes the nature of the solution of the system of equations corresponding to the following augmented matrix?
⌊ | ⌋ 1 0 1 2 | 4 || 0 1 3 - 1| 2 || ⌈ 0 1 0 - 4|- 7 ⌉ 0 0 1 1 3
a) The solution is unique.
b) There are infinitely many solutions.
c) The system is inconsistent, there are no solutions.
d) There is not enough information to decide.

 

Not correct. Choice (a) is false.
Your answer is correct.
Reduce the matrix to row echelon form.
⌊ | ⌋ 1 0 1 2 | 4 || 0 1 3 - 1 | 2 || ⌈ 0 1 0 - 4 |- 7 ⌉ 0 0 1 1 3
R--3-:=R-3----R-→2⌊ | ⌋ 1 0 1 2 | 4 || 0 1 3 - 1 | 2 || ⌈ 0 0 - 3 - 3 |- 9 ⌉ 0 0 1 1 3
   
1 R--3-:=-R3----3-R-2→⌊ | ⌋ 1 0 1 2|4 || 0 1 3 - 1|2 || ⌈ 0 0 1 1|3 ⌉ 0 0 1 1 3
   
-R-4-:=-R-4---R-3→⌊ | ⌋ 1 0 1 2 |4 || 0 1 3 - 1 |2 || ⌈ 0 0 1 1 |3 ⌉ 0 0 0 0 0
 
The matrix is consistent and the fourth column does not have a leading one, so the system has infinitely many solutions.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
It is always possible to decide such questions by reducing the augmented matrix to row echelon form.
 

Question 9

 
 
Which is the correct solution for the system of equations corresponding to the augmented matrix
⌊ ⌋ 1 2 1 1 ⌈ 0 1 1 2 ⌉? 0 0 1 4
a) x = 3, y = -3, z = 4   b) x = -1, y = -1, z = 4
c) x = 1, y = -2, z = 4   d) x = -3, y = 0, z = 4

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Using back substitution we see that
z = 4 y = 2- z = 2- 4 = - 2 x = - 2y - z + 1 = 4- 4+ 1 = 1.
Not correct. Choice (d) is false.
 

Question 10

 
 
Which of the statements below correctly describe the relationship between the following matrices?
[ ] [ ] A = 1 0 1 , B = 1 0 2 , 0 1 1 0 1 1 [ ] [ ] C = 1 1 2 , D = 1 2 4 . 0 1 1 0 1 1
(More than one answer may be correct.)
a) A is row equivalent to B   b) A is row equivalent to C
c) B is row equivalent to D.   d) A is row equivalent to D
e) B is row equivalent to C.

 

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 true.
A is row equivalent to C because
[ ]R :=R -R [ ] C = 1 1 2 -1----1---2→ 1 0 1 = A 0 1 1 0 1 1
There is at least one mistake.
For example, choice (c) should be true.
B is row equivalent to D because
[ ] [ ] D = 1 2 4 -R-1-:=-R-1----2R-2→ 1 0 2 = B 0 1 1 0 1 1
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 false.
Your answers are correct
  1. False.
  2. True. A is row equivalent to C because
    [ ]R :=R -R [ ] C = 1 1 2 -1----1---2→ 1 0 1 = A 0 1 1 0 1 1
  3. True. B is row equivalent to D because
    [ ] [ ] D = 1 2 4 -R-1-:=-R-1----2R-2→ 1 0 2 = B 0 1 1 0 1 1
  4. False.
  5. False.