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Quiz 7: Matrix Operations

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.
Try again. The size of a matrix is the number of rows × the number of columns.
Not correct. Choice (b) is false.
Try again. The size of a matrix is the number of rows × the number of columns.
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.
Try again. The size of a matrix is the number of rows × the number of columns.
 

Question 2

 
 
What is the (2,3) entry in 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.
Try again. The (2,3) entry is in row 2 and column 3.
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.
Try again. The (2,3) entry is in row 2 and column 3.
Not correct. Choice (d) is false.
Try again. The (2,3) entry is in row 2 and column 3.
 

Question 3

 
 
Which sums can be made from the following matrices ?
[ ] [ ] ⌊ 1 2 3⌋ [ ] A = 1 2 3 ,B = 1 2 ,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 true.
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 true.
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. True.
  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. True.
 

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

 
 
If A = ⌊ ⌋ ⌈1 0 3⌉ 2 - 1 2 0 2 1 find the matrix 7A.
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 3 ⌈1 2⌉ ⌈1 2 3⌉ 1 2 A = 4 5 6 , B = 3 4 , C = 4 5 6 and D = 3 4 . 5 6 7 8 9
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

 
 
If A is a 2 × 3 matrix, and B is a 3 × 4 matrix, how many columns does AB have? (Enter your answer into the answer box.)

 

Your answer is correct
AB is a 2 × 4 matrix, and so has 4 columns.
Not correct. You may try again.
Remember that an n×m matrix, multiplied by an m×p matrix, is an n×p matrix.
 

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 = [ ] 3 - 10 - 2 - 7.
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 compute the square of either of the two matrices Y and W.
There is at least one mistake.
For example, choice (e) should be false.
Y W - Z2 = [ ] 3 - 10 - 2 - 7.
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 = [ ] 3 - 10 - 2 - 7.
  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 compute the square of either of the two matrices Y and W.
  5. False. Y W - Z2 = [ ] 3 - 10 - 2 - 7.
  6. True.
  7. True.