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Quiz 11: Determinants

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Find the determinant of the matrix A = ⌊ ⌋ 1 - 3 - 4 ⌈ 2 1 0⌉ 3 - 2 5. (Enter your answer into the answer box.)

 

Your answer is correct

Not correct. You may try again.
Expanding by the second row (for example),

| | | | det(A) = - 2||- 3 - 4||+ ||1 - 4||. |- 2 5 | |3 5|
 

Question 2

 
 
Find the determinant of the matrix A = ⌊ 1 7 6 ⌋ ⌈ 5 1 - 2⌉ 3 8 4. (Enter your answer into the answer box.)

 

Your answer is correct

Not correct. You may try again.
Expanding by the first column (for example),

| | | | | | ||1 - 2|| ||7 6|| ||7 6 || det(A ) = |8 4 |- 5|8 4|+ 3|1 - 2|.
 

Question 3

 
 
Let A = ⌊ ⌋ ⌈1 2 3 ⌉ 1 0 - 1 3 4 5. Which of the following statements is correct ?
a) A is invertible since det A = 0   b) A is not invertible since det A = 0
c) A is invertible since det A⁄=0   d) A is not invertible since det A⁄=0

 

Not correct. Choice (a) is false.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 4

 
 
Evaluate ||1 - 3 - 4|| ||2 1 0|| ||2 - 6 - 8||. (Enter your answer into the answer box.)

 

Your answer is correct
Since R3 = 2 × R1 the determinant is zero.
Not correct. You may try again.
Note that R3 = 2 × R1.
 

Question 5

 
 
Suppose that A is a 3 × 3 matrix and det (A) = -3. What is det (4A)?
a) -3 4   b) -3
c) -12   d) -192

 

Not correct. Choice (a) is false.
4A is obtained by multiplying every entry in A by 4. Remember that if one row (or column) of A is multiplied by 4 then the determinant of the resulting matrix is equal to 4 times det (A).
Not correct. Choice (b) is false.
4A is obtained by multiplying every entry in A by 4. Remember that if one row (or column) of A is multiplied by 4 then the determinant of the resulting matrix is equal to 4 times det (A).
Not correct. Choice (c) is false.
4A is obtained by multiplying every entry in A by 4. Remember that if one row (or column) of A is multiplied by 4 then the determinant of the resulting matrix is equal to 4 times det (A).
Your answer is correct.
 

Question 6

 
 
Evaluate det (A) if A = ⌊ ⌋ 5 - 3 - 4 ⌈0 2 6 ⌉ 0 0 9. (Enter your answer into the answer box.)

 

Your answer is correct
The product of the diagonal entries = 90.
Not correct. You may try again.
Since A is triangular, the determinant is the product of the diagonal entries.
 

Question 7

 
 
Let A be a 3 × 3 matrix, and suppose that the matrix B is obtained from A by the following row operations:
R1 ↔ R2; R3 → R3+ 5R2
If det (A) = 12, what is det (B)?
a) -60   b) -12
c) 12   d) 60

 

Not correct. Choice (a) is false.
Adding a multiple of a row to another row does not change the value of the determinant.
Your answer is correct.
Not correct. Choice (c) is false.
Swapping rows changes the sign of the determinant.
Not correct. Choice (d) is false.
Adding a multiple of a row to another row does not change the value of the determinant.
 

Question 8

 
 
If A = ⌊ ⌋ ⌈3 - 1 9⌉ 0 2 6 0 0 5, what is det (7A)?
a) 30   b) 210
c) 1470   d) 10290

 

Not correct. Choice (a) is false.
det (7A) = 73 det (A).
Not correct. Choice (b) is false.
det (7A) = 73 det (A).
Not correct. Choice (c) is false.
det (7A) = 73 det (A).
Your answer is correct.
 

Question 9

 
 
Let A = ⌊ ⌋ k k 4 ⌈0 k 5⌉ 1 1 k. Which one of the following statements is true?
a) If k = 0 then A is invertible.
b) A is invertible if k is any real number other than 0 or 4.
c) A is invertible if k = 0 or 4.
d) A is invertible if k is any real number other than 0, 2 or -2.
e) A is invertible if k = 0, 2 or -2.

 

Not correct. Choice (a) is false.
A is invertible if det (A) ⁄= 0.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
A is invertible if det (A) ⁄= 0.
Your answer is correct.
Not correct. Choice (e) is false.
A is invertible if det (A) ⁄= 0.
 

Question 10

 
 
Let A = ⌊ ⌋ - 2 1 4 ⌈ 0 3 7⌉ 0 0 1. Find any values of λ such that det (A - λI) = 0.
a) λ = -2, 0   b) λ = 0, 1
c) λ = -2, 1, 4   d) λ = -2, 1, 3
e) λ = 0, 1, 3   f) λ = 1, 4, 7
g) λ = 0, 3, 7

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.
Not correct. Choice (f) is false.
Not correct. Choice (g) is false.