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Quiz 10: Determinants

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Question 1

Let A = ⌊ ⌋ 1 - 3 - 4 ⌈2 1 0 ⌉ 3 - 2 5. Which of the following is true ?

a)
det(A) = 3;
   b)
det(A) = 63;
c)
det(A) = 7;
   d)
det(A) is undefined;
e)
none of the above.

 

Not correct. Choice (a) is false.
Your answer is correct.
The determinant of a square matrix is always defined, so (4) is wrong.
Either:
det(A) = | | ||1 - 3 - 4|| ||2 1 0|| |3 - 2 5|
= 1||1 0|| ||2 1||- (-3)||2 0|| ||3 5|| + (-4)||2 1|| ||3 - 2||
= 1(1 × 5 - 0 × 3) - (-3)(2 × 5 - 0 × 3) + (-4)(2 × (-2) - 1 × 3)
= 1 × 5 - (-3) × 10 + (-4) × (-7) = 63,
or, using elementary row operations R2 := R2 - 2R1;R3 := R3 - 3R1R3 := R3 -R2 (all of which give matrices with the same determinant) to reduce to (upper) triangular form:
||1 - 3 - 4|| ||1 - 3 - 4|| ||1 - 3 - 4|| det(A ) = ||0 7 8|| = ||0 7 8 ||= ||0 7 8|| = 1× 7× 9 = 63. ||3 - 2 5|| ||0 7 17|| ||0 0 9||
Hence the correct answer is (2).
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

Question 2

Let A = ⌊ ⌋ 1 - 3 - 4 0 ⌈2 1 0 0⌉ 3 - 2 5 0. Which of the following is true ?

a)
det(A) = 63;
   b)
det(A) = 7;
c)
det(A) = 0;
   d)
det(A) is undefined;
e)
none of the above.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
The determinant of a non-square matrix is undefined, so (4) is correct.
Not correct. Choice (e) is false.

Question 3

Let A = ⌊1 2 3 4 5 6⌋ |7 8 9 1 2 0| ||3 4 5 6 0 0|| ||7 8 9 0 0 0|| |⌈1 2 0 0 0 0|⌉ 3 0 0 0 0 0. Which of the following is true ?

a)
det(A) = 3888;
   b)
det(A) = 0;
c)
det(A) = 28;
   d)
det(A) is undefined;
e)
none of the above.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Your answer is correct.
The determinant of a square matrix is always defined, so (4) is wrong.
A is not triangular (which means all zero on one side of the main top-left to bottom-right diagonal, not the top-right to bottom-left diagonal), so det(A) is not necessarily the product of the “wrong” diagonal elements, i.e., not necessarily 6 ⋅ 2 ⋅ 6 ⋅ 9 ⋅ 2 ⋅ 3 = 3888.
The three elementary row operations R1 ↔ R6; R2 ↔ R5; R3 ↔ R4; change the sign of the determinant three times, i.e.,
| | ||3 0 0 0 0 0|| ||1 2 0 0 0 0|| det(A ) = - ||7 8 9 0 0 0||= - 3 ⋅2⋅9⋅6 ⋅2⋅6 = - 3888. ||3 4 5 6 0 0|| ||7 8 9 1 2 0|| |1 2 3 4 5 6|
So the correct answer is (5).

Question 4

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.
| | | | | | ||0 - 1|| ||1 - 1|| ||1 0|| det A = 1|4 5|- 2 |3 5| +3 |3 4| = 4- 16 + 12 = 0.
It is not invertible since det A = 0.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 5

What is the determinant of the matrix ⌊ 3 2 1 - 2⌋ | 1 1 2 3 | |⌈ 0 1 0 - 1|⌉ 2 3 4 5 ?

 

Your answer is correct
|| || ||3 2 1 - 2|| ||1 1 2 3|| ||0 1 0 - 1|| 2 3 4 5 = -|| || ||0 1 0 - 1|| ||1 1 2 3 || ||3 2 - 1 2 || 2 3 4 5
= -( | | | |) ||1 2 3|| ||1 1 2 || ( - 1||3 - 1 2||+ 1||3 2 - 1||) |2 4 5| |2 3 4 |
= -( ||- 1 2|| ||3 2|| ||3 - 1|| ||2 - 1|| ||3 - 1|| ||3 2||) - ||4 5||+ 2||2 5||- 3 ||2 4|| +||3 4||- ||2 4||+ 2||2 3||
= -(+13 + 22 - 42 + 11 - 14 + 10)
= 0.

Not correct. You may try again.
Try expanding the determinant along the third row.

Question 6

Let A = ⌊ ⌋ a b c ⌈d e f⌉ g h i. Which of the following is true ?

a)
det(A) = aei + afh + bdi + bgf + cdh + ceg;
b)
det(A) = aei - afh - bdi + bgf + cdh - ceg;
c)
det(A) = aei + afh + bdi - bgf - cdh - ceg;
d)
det(A) = aei - afh + bdi - bgf + cdh - ceg;
e)
none of the above.

 

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

det(A) = || || ||a b c|| ||d e f|| g h i
= a| | ||e f|| |h i|- b| | ||d f || |g i| + c| | ||d e|| |g h|
= a(ei - fh) - b(di - fg) + c(dh - eg)
= aei - afh + bdi - bgf + cdh - ceg.
Hence the correct answer is (4).
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

Question 7

The matrix A has the elementary row operations

R2 := R2 - R1, R2 ↔ R3, R3 := R3 - 3R2,
applied to it to transform it to the matrix ⌊ ⌋ 1 - 1 1 ⌈0 1 - 1⌉ 0 0 6. What is the determinant of A ?
a)
-6
   b)
6
c)
18
   d)
-18
e)
None of the above

 

Your answer is correct.
The determinant of A has the opposite sign of the determinant of the matrix
⌊ ⌋ ⌈1 - 1 1⌉ 0 1 - 1 0 0 6
since a row swap has been applied. The determinant of a triangular matrix is the product of the diagonal entries hence detA = -1 × 1 × 6 = -6.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

Question 8

The matrix A has the elementary row operations

R2 := R2 - 3R1,R3 := R3 + 4R1,R2 ↔ R3,R2 := 12R2,R4 := R4 - 2R2,
applied to it to transform it to the matrix ⌊ ⌋ 2 1 0 3 ||0 1 2 1|| ⌈0 0 3 4⌉ 0 0 0 1. What is the determinant of A ?
a)
-3
   b)
3
c)
12
   d)
-12
e)
None of the above

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Call the new matrix A′. Now detA′ = -1 2 detA since A′ has had a row swap to change the sign and A′ has had a row divided by 2. Now detA′ = 6 since it is triangular. Thus detA = -2 × 6 = -12.
Not correct. Choice (e) is false.

Question 9

Let A = ⌊a b c⌋ ⌈d e f⌉ g h i, and suppose det(A) = 0. Consider the system

ax + by + cz = 5
dx + ey + fz = 1
gx + hy + iz = 4
Which of the following is necessarily true ?
a)
the system has no solutions;
   b)
the system has many solutions;
c)
the system has a unique solution ;
   d)
none of the above.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Since det(A) = 0, A is not invertible, so we cannot solve the system by left multiplying by A-1 both sides of AxT = bT (x = [xyz], b = [514]) to get a unique solution xT = A-1bT.
In fact, if we solve by reducing the augmented matrix [A|bT], the reduced form will have third row [000|*]. Hence, at least one column will be missing a leading 1, so (c) cannot possibly be true.
If *≠0, (a) will be true. If * = 0, (b) will be true.
However we do not have enough information to decide if * = 0 or not. So neither (a) nor (b) is necessarily true. Hence the correct answer is (d)!!

Question 10

Which of the following statements about determinants are true ?
 

a)
detA-1 = -detA
  b)
detA-1 = 1 detA-
c)
| | ||a b c|| ||d e f|| |g h i| = | | ||a d g|| ||b e h|| |c f i|
   d)
| | ||2 4 6 || ||8 10 12|| |14 16 18| = 2| | ||1 2 3|| ||4 5 6|| |7 8 9|

 

Not correct. Choice (a) is false.
1 = detI = det(AA-1) = detAdetA-1.
Your answer is correct.
1 = detI = det(AA-1) = detAdetA-1.
Your answer is correct.
This is correct because detAT = detA.
Not correct. Choice (d) is false.
As | | ||2 4 6 || ||8 10 12|| |14 16 18| = 23| | ||1 2 3|| ||4 5 6|| |7 8 9|, this is incorrect.
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