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Quiz 7: Sums and products of matrices

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
What is the size of the matrix  ⌊ ⌋ 1 2 3 4 A = ⌈ 12 3 4 1⌉ 13 14 1 2  ?
a) 2 × 3   b) 3 × 2
c) 3 × 4   d) 4 × 3

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
A has 3 rows and 4 columns and hence it is a 3 × 4 matrix.
Not correct. Choice (d) is false.
 

Question 2

 
 
What is entry (2,3) of the matrix ⌊ ⌋ 1 2 3 4 A = ⌈12 3 4 1⌉ 13 14 1 2  ?
a) 3   b) 4
c) 12   d) 14

 

Not correct. Choice (a) is false.
Your answer is correct.
We are looking for the entry in the second row and the third column of A, which is 4.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 3

 
 
Which sums can be made from the following matrices ?
⌊ ⌋ ⌊ ⌋ [ ] 1 2 1 2 3 [ ] A = 1 2 3 ,B = ⌈3 4⌉ ,C = ⌈4 5 6⌉,D = 1 2 . 4 5 6 5 6 7 8 9 3 4
a) A + A   b) A + B   c) A + C
d) A + D   e) B + A   f) B + B
g) B + D   h) C + A   i) C + C
j) C + D   k) D + A   l) D + C
m) D + D   n) D + B

 

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.
Only matrices of the same size can be added.
There is at least one mistake.
For example, choice (c) should be false.
Only matrices of the same size can be added.
There is at least one mistake.
For example, choice (d) should be false.
Only matrices of the same size can be added.
There is at least one mistake.
For example, choice (e) should be false.
Only matrices of the same size can be added.
There is at least one mistake.
For example, choice (f) should be true.
There is at least one mistake.
For example, choice (g) should be false.
Only matrices of the same size can be added.
There is at least one mistake.
For example, choice (h) should be false.
Only matrices of the same size can be added.
There is at least one mistake.
For example, choice (i) should be true.
There is at least one mistake.
For example, choice (j) should be false.
Only matrices of the same size can be added.
There is at least one mistake.
For example, choice (k) should be false.
There is at least one mistake.
For example, choice (l) should be false.
Only matrices of the same size can be added.
There is at least one mistake.
For example, choice (m) should be true.
There is at least one mistake.
For example, choice (n) should be false.
Only matrices of the same size can be added.
Your answers are correct
  1. True.
  2. False. Only matrices of the same size can be added.
  3. False. Only matrices of the same size can be added.
  4. False. Only matrices of the same size can be added.
  5. False. Only matrices of the same size can be added.
  6. True.
  7. False. Only matrices of the same size can be added.
  8. False. Only matrices of the same size can be added.
  9. True.
  10. False. Only matrices of the same size can be added.
  11. False.
  12. False. Only matrices of the same size can be added.
  13. True.
  14. False. Only matrices of the same size can be added.
 

Question 4

 
 
Let A = [1 - 3 4] 2 10 - 7 and B = [5 - 2 - 3] 0 11 12. What is A + B?
a) [6 - 5 - 1] 2 21 5   b) [ ] 1 - 3 4 5 - 2 - 3 2 10 - 7 0 11 12
c) [6 - 5 1] 2 21 5   d) ⌊1 - 3 4⌋ |2 10 - 7| |⌈5 - 2 - 3|⌉ 5 - 2 - 3
e) [6 5 1] 2 21 5   f) A + B is undefined

 

Not correct. Choice (a) is false.
Matrix addition is defined by adding the corresponding entries of the two matrices.
Not correct. Choice (b) is false.
Matrix addition is defined by adding the corresponding entries of the two matrices.
Your answer is correct.
Not correct. Choice (d) is false.
Matrix addition is defined by adding the corresponding entries of the two matrices.
Not correct. Choice (e) is false.
Not correct. Choice (f) is false.
Matrix addition is defined by adding the corresponding entries of the two matrices.
 

Question 5

 
 
What is the matrix 7Z if Z = ⌊ ⌋ ⌈1 0 3⌉ 2 - 1 2 0 2 1?
a) ⌊7 0 21⌋ ⌈2 - 1 2 ⌉ 0 2 1   b) ⌊1 0 3 ⌋ ⌈14 - 7 14⌉ 0 2 1
c) ⌊ ⌋ 1 0 3 ⌈2 - 1 2⌉ 0 14 7   d) ⌊ ⌋ 7 0 21 ⌈14 - 7 14⌉ 0 14 7

 

Not correct. Choice (a) is false.
When multiplying a matrix by a scalar, each entry in the matrix is multiplied by the scalar.
Not correct. Choice (b) is false.
When multiplying a matrix by a scalar, each entry in the matrix is multiplied by the scalar.
Not correct. Choice (c) is false.
When multiplying a matrix by a scalar, each entry in the matrix is multiplied by the scalar.
Your answer is correct.
 

Question 6

 
 
Consider the four matrices
⌊ ⌋ ⌊ ⌋ [ ] 1 2 1 2 3 [ ] A = 1 2 3 ,B = ⌈3 4⌉ ,C = ⌈4 5 6⌉andD = 1 2 . 4 5 6 5 6 7 8 9 3 4
Which of the following products can be made from these matrices ?
a) AB   b) AC   c) AD
d) A2   e) BA   f) BD
g) B2   h) CA   i) CB
j) CD   k) CB   l) C2
m) DA   n) DC   o) CB
p) D2

 

There is at least one mistake.
For example, choice (a) should be true.
A is a 2 × 3 matrix hence we can only post-multiply A by a matrix with 3 rows and pre-multiply A by a matrix with 2 columns.
There is at least one mistake.
For example, choice (b) should be true.
There is at least one mistake.
For example, choice (c) should be false.
A is a 2 × 3 matrix hence we can only post-multiply A by a matrix with 3 rows and pre-multiply A by a matrix with 2 columns.
There is at least one mistake.
For example, choice (d) should be false.
A is a 2 × 3 matrix hence we can only post-multiply A by a matrix with 3 rows and pre-multiply A by a matrix with 2 columns.
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.
There is at least one mistake.
For example, choice (g) should be false.
B is a 3 × 2 matrix hence we can only post-multiply B by a matrix with 2 rows and pre-multiply B by a matrix with 3 columns.
There is at least one mistake.
For example, choice (h) should be false.
C is a 3 × 3 matrix hence we can only post-multiply C by a matrix with 3 rows and pre-multiply C by a matrix with 3 columns
There is at least one mistake.
For example, choice (i) should be true.
There is at least one mistake.
For example, choice (j) should be false.
C is a 3 × 3 matrix hence we can only post-multiply C by a matrix with 3 rows and pre-multiply C by a matrix with 3 columns
There is at least one mistake.
For example, choice (k) should be true.
There is at least one mistake.
For example, choice (l) should be true.
There is at least one mistake.
For example, choice (m) should be true.
There is at least one mistake.
For example, choice (n) should be false.
C is a 3 × 3 matrix hence we can only post-multiply C by a matrix with 3 rows and pre-multiply C by a matrix with 3 columns
There is at least one mistake.
For example, choice (o) should be true.
There is at least one mistake.
For example, choice (p) should be true.
Your answers are correct
  1. True. A is a 2 × 3 matrix hence we can only post-multiply A by a matrix with 3 rows and pre-multiply A by a matrix with 2 columns.
  2. True.
  3. False. A is a 2 × 3 matrix hence we can only post-multiply A by a matrix with 3 rows and pre-multiply A by a matrix with 2 columns.
  4. False. A is a 2 × 3 matrix hence we can only post-multiply A by a matrix with 3 rows and pre-multiply A by a matrix with 2 columns.
  5. True.
  6. True.
  7. False. B is a 3 × 2 matrix hence we can only post-multiply B by a matrix with 2 rows and pre-multiply B by a matrix with 3 columns.
  8. False. C is a 3 × 3 matrix hence we can only post-multiply C by a matrix with 3 rows and pre-multiply C by a matrix with 3 columns
  9. True.
  10. False. C is a 3 × 3 matrix hence we can only post-multiply C by a matrix with 3 rows and pre-multiply C by a matrix with 3 columns
  11. True.
  12. True.
  13. True.
  14. False. C is a 3 × 3 matrix hence we can only post-multiply C by a matrix with 3 rows and pre-multiply C by a matrix with 3 columns
  15. True.
  16. True.
 

Question 7

 
 
Which of the following matrices is the identity matrix I4 ?
a) [4 0] 0 4   b) ⌊1 0 0 0⌋ |0 1 0 0| |⌈0 0 1 0|⌉ 0 0 0 1
c) [1 0] 0 1   d) ⌊ ⌋ 1 0 0 ⌈0 1 0⌉ 0 0 1

 

Not correct. Choice (a) is false.
This is the matrix 4I2.
Your answer is correct.
The In matrix is an n × n matrix with 1’s down the diagonal and zeros elsewhere.
Not correct. Choice (c) is false.
This is the matrix I2.
Not correct. Choice (d) is false.
This is the matrix I3.
 

Question 8

 
 
Let A = ⌊ ⌋ 1 - 3 4 ||2 10 - 7|| ⌈0 1 0⌉ 0 0 0 and B = ⌊ 5 0⌋ ⌈- 2 11⌉ - 3 12. Which of the following matrices is equal to the matrix AB?
a) ⌊ ⌋ |- 1 15| |⌈ 11 26|⌉ - 3 12 0 0   b) ⌊ ⌋ - 1 0 ||- 4 26|| ⌈ 0 12⌉ 0 0
c) ⌊ ⌋ - 1 15 ||11 26|| ⌈- 2 11⌉ 0 0   d) ⌊ ⌋ - 1 15 ||11 26|| ⌈- 2 11⌉ 6 0
e) AB is undefined

 

Not correct. Choice (a) is false.
Since A is a 4 × 3 matrix and B is a 3 × 2 matrix the product AB is a 4 × 2 matrix.
Not correct. Choice (b) is false.
The entry in row 1 and column 2 of AB is 15.
Your answer is correct.
Not correct. Choice (d) is false.
The entry in row 4 and column 1 of AB is 0.
Not correct. Choice (e) is false.
Since A is a 4 × 3 matrix and B is a 3 × 2 matrix the product AB is a 4 × 2 matrix. In particular, the product of A and B is defined.
 

Question 9

 
 
Let
⌊ ⌋ [ ] 1 0 [ ] W = 1 2 3 , X = ⌈2 - 1⌉ and Y = 0 1 3 2 1 0 2 - 2 5
Which of the following statements are correct?
a) WX + Y = [ ] 5 5 5 5   b) XY = ⌊ 0 1 ⌋ ⌈ 2 - 3⌉ - 4 10
c) XW + Y = [ ] 2 5 3 0   d) Y W = [ ] 3 2 1 13 6 - 1
e) WXY = [ ] - 8 20 - 8 13

 

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 true.
There is at least one mistake.
For example, choice (c) should be false.
XW is a 3 × 3 matrix so the sum XW + Y is does not make sense because we can only add matrices of the same size.
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.
WX = [ ] 5 4 - 7 4 so WXY = [ ] - 8 25 - 8 13.
Your answers are correct
  1. True.
  2. True.
  3. False. XW is a 3 × 3 matrix so the sum XW + Y is does not make sense because we can only add matrices of the same size.
  4. True.
  5. False. WX = [ ] 5 4 - 7 4 so WXY = [ ] - 8 25 - 8 13.
 

Question 10

 
 
Let
⌊ ⌋ ⌊ ⌋ [ ] [ ] ⌈1 - 3⌉ ⌈ 3 2 1⌉ 1 2 0 0 1 W = 2 - 2 ,X = - 1 2 3 , Y = 0 - 1 2 and Z = 2 3 . 3 1 2 - 1 1
Which of the following statements are true ?
a) WY - X = ⌊ - 2 3 - 7⌋ ⌈ 3 4 - 7⌉ 1 6 1   b) Y W - Z2 = [ ] 5 - 9 0 - 2
c) The product (Y W)2 is not defined.   d) (Y W)2 = Y 2W2
e) Y W - Z2 = [ ] 5 - 8 2 1   f) WY - X = ⌊ ⌋ - 2 3 - 7 ⌈ 3 4 - 7⌉ 1 6 1,
g) Y W - Z2 = [ ] 3 - 10 - 2 - 7.

 

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.
Y W - Z2 = [ ] 5 - 8 2 1
There is at least one mistake.
For example, choice (c) should be false.
Y W is a 2 × 2 matrix so its square is defined. In fact, (Y W)2 = [ ] - 3 - 63 36 - 12
There is at least one mistake.
For example, choice (d) should be false.
If A is a matrix then A2 is defined if and only if A is a square matrix. Consequently, we cannot copmute the square of the either of the two matrices Y and W.
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.
There is at least one mistake.
For example, choice (g) should be true.
Your answers are correct
  1. True.
  2. False. Y W - Z2 = [ ] 5 - 8 2 1
  3. False. Y W is a 2 × 2 matrix so its square is defined. In fact, (Y W)2 = [ ] - 3 - 63 36 - 12
  4. False. If A is a matrix then A2 is defined if and only if A is a square matrix. Consequently, we cannot copmute the square of the either of the two matrices Y and W.
  5. True.
  6. True.
  7. True.