menuicon

MATH1002 Quizzes

Quiz 10: Determinants
Question 1 Questions
Let A = 134 2 1 0 32 5 . Which of the following is true ? Exactly one option must be correct)
a)
det(A) = 3;
b)
det(A) = 63;
c)
det(A) = 7;
d)
det(A) is undefined;
e)
none of the above.

Choice (a) is incorrect
Choice (b) is correct!
The determinant of a square matrix is always defined, so (4) is wrong.
Either: det(A) = 134 2 1 0 32 5 = 1 10 2 1 (3) 20 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: det(A) = 134 0 7 8 32 5 = 134 0 7 8 0 7 17 = 134 0 7 8 0 0 9 = 1×7×9 = 63. Hence the correct answer is (2).
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Let A = 1340 2 1 0 0 32 5 0 . Which of the following is true ? Exactly one option must be correct)
a)
det(A) = 63;
b)
det(A) = 7;
c)
det(A) = 0;
d)
det(A) is undefined;
e)
none of the above.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The determinant of a non-square matrix is undefined, so (4) is correct.
Choice (e) is incorrect
Let A = 123456 7 8 9 1 2 0 345600 7 8 9 0 0 0 120000 3 0 0 0 0 0 . Which of the following is true ? Exactly one option must be correct)
a)
det(A) = 3888;
b)
det(A) = 0;
c)
det(A) = 28;
d)
det(A) is undefined;
e)
none of the above.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) 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., det(A) = 300000 1 2 0 0 0 0 789000 3 4 5 6 0 0 789120 1 2 3 4 5 6 = 329626 = 3888. So the correct answer is (5).
Let A = 12 3 1 0 1 34 5 . Which of the following statements is correct ? Exactly one option must be correct)
a)
A is invertible since det A = 0
b)
A is not invertible since det A = 0
c)
A is invertible since det A0
d)
A is not invertible since det A0

Choice (a) is incorrect
Choice (b) is correct!
detA = 1 01 4 5 2 11 3 5 + 3 10 3 4 = 4 16 + 12 = 0. It is not invertible since det A = 0.
Choice (c) is incorrect
Choice (d) is incorrect
What is the determinant of the matrix 3212 1 1 2 3 0101 2 3 4 5 ?

Correct!
3212 1 1 2 3 0101 2 3 4 5 = 01 0 1 1 1 2 3 321 2 2 3 4 5 = 1 1 2 3 3 1 2 2 4 5 + 1 11 2 3 2 1 23 4 = 12 4 5 + 2 32 2 5 3 31 2 4 + 21 3 4 31 2 4 + 2 32 2 3 = (+13 + 22 42 + 11 14 + 10) = 0.

Incorrect. Please try again.
Try expanding the determinant along the third row.
Let A = abc d e f g hi . Which of the following is true ? Exactly one option must be correct)
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.

Choice (a) is incorrect
Choice (b) is correct!

det(A) = abc d e f g hi = a ef hi b df g i + c de g h = a(ei fh) b(di fg) + c(dh eg) = aei afh + bdi bgf + cdh ceg. Hence the correct answer is (4).
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
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 11 1 0 1 1 0 0 6 . What is the determinant of A ? Exactly one option must be correct)
a)
-6
b)
6
c)
18
d)
-18
e)
None of the above

Choice (a) is correct!
The determinant of A has the opposite sign of the determinant of the matrix
11 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.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
The matrix A has the elementary row operations
R2 := R2 3R1,R3 := R3 + 4R1,R2 R3,R2 := 1 2R2,R4 := R4 2R2,
applied to it to transform it to the matrix 2103 0 1 2 1 0034 0 0 0 1 . What is the determinant of A ? Exactly one option must be correct)
a)
-3
b)
3
c)
12
d)
-12
e)
None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Call the new matrix A. Now detA = 1 2detA 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.
Choice (e) is incorrect
Let A = abc d e f g hi , 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 ? Exactly one option must be correct)
a)
the system has no solutions;
b)
the system has many solutions;
c)
the system has a unique solution ;
d)
none of the above.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Since det(A) = 0, A is not invertible, so we cannot solve the system by left multiplying by A1 both sides of AxT = bT (x = [xyz], b = [514]) to get a unique solution xT = A1bT.
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)!!
Which of the following statements about determinants are true ?
 
Exactly one option must be correct)
a)
detA1 = detA
b)
detA1 = 1 detA
c)
abc d e f g hi = adg b eh c f i
d)
2 4 6 8 10 12 141618 = 2 123 4 5 6 789

Choice (a) is incorrect
1 = detI = det(AA1) = detAdetA1.
Choice (b) is correct!
1 = detI = det(AA1) = detAdetA1.
Choice (c) is correct!
This is correct because detAT = detA.
Choice (d) is incorrect
As 2 4 6 8 10 12 141618 = 23 123 4 5 6 789 , this is incorrect.