Quiz 1
Maximum marks: 10 Due date: 10th January
You need to submit answers through a quiz assignment that I will share on Google
classroom.
Submissions from non-IITH email ID accounts will be awarded zero marks.
Honor code (cheating policy): These problems are designed to reinforce the material
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being taught. You are welcome to discuss them with your classmates or use any books
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and online resources. However, my expectation is that you do not simply copy and paste.
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Before submitting your work, ensure that you fully understand the underlying concepts.
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1. Consider the following matrices.
1 1 1 1 0 2 0
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(i)
0 0 0 (iii) 0 1 3 0
0 0 0 1
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1 1 1 0 0 1
(ii) (iv) .
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0 1 0 0 1 0
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Which of the above matrices are NOT in row-reduced echelon matrix form.
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(a) (i) and (ii) (b) (i) and (iii) (c) (ii) and (iv) (d) (iii) and (iv)
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2. Which of the following statements is FALSE about elementary row operations:
(a) It preserves the size of the matrix operated on.
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(b) It preserves the number of non-zero rows.
(c) It preserves the number of non-zero columns.
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(d) It preserves the solution set of the corresponding linear system.
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3. The augmented matrix A′ of the system of equation AX = b is row equivalent to the following matrix
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0 1 0 2 3 4 −17
0 0 1 0 0 4 8
0 0 0 0 1 4 24 .
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0 0 0 0 0 0 4
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Which of the following options about the system AX = b is correct.
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(a) It has no solutions. (b) It has a unique solution. (c) It has infinitely many solu-
tions.
4. The augmented matrix A′ of the system of equation AX = b is row equivalent to the following matrix
0 1 0 2 3 0 −17
0 0 1 0 0 0 8
0 0 0 0 1 2 24 .
0 0 0 0 0 0 0
Which of the following options about the system AX = b is correct.
(a) It has no solutions. (b) It has a unique solution. (c) It has infinitely many solu-
tions.
5. The augmented matrix A′ of the system of equation AX = b is row equivalent to the following matrix
0 1 0 2 0 0 −17
0 0 1 0 0 0 8
.
0 0 0 0 1 3 1
0 0 0 0 0 0 0
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Which of the following options about the system AX = b is correct.
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(a) It has infinitely many solutions with x1 , x6 as free variables.
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(b) It has infinitely many solutions with x1 , x4 as free variables.
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(c) It has infinitely many solutions with x4 , x6 as free variables.
(d) It has infinitely many solutions with x1 , x4 , x6 as free variables.
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6. Let C3×3 be the set of 3 × 3 matrices over C. Define a binary operation ∗ as
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A ∗ B = A · B − B · A,
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where for any X, Y ∈ C3×3 , X · Y means the usual matrix product. Which of the following options
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about the operation ∗ is correct.
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(a) ∗ is an associatative operation.
(b) ∗ is a commutative operation.
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(c) A ∗ (B ∗ C) + B ∗ (C ∗ A) + C ∗ (A ∗ B) = 0 for all A, B, C ∈ C3×3 .
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7. Which of the following statements is FALSE:
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(a) C is a vector space over R (with usual operations).
(b) C is a vector space over Q (with usual operations).
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(c) R is a vector space over Q (with usual operations).
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(d) Q is a vector space over R (with usual operations).
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8. Let V = C (with usual addition) and let F = C. Let λ ∗ v denotes the scalar multiplication of λ and v.
Which of the following definitions of ∗ makes V a vector space over F.
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(a) λ ∗ v = λ2 v.
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(b) λ ∗ v = Re(λ)v, where Re(λ) denotes the real value of λ.
(c) λ ∗ v = λv, where λ denotes the complex conjugate of λ.
(d) λ ∗ v = (λ + 1)v.
9. Let F = C and let V be the set of nonsingular 3 × 3 matrices with complex entries. Define vector
addition ∗ by matrix addition, and scalar multiplication as (λA)ij = λAij for all i, j. Then which of the
following is true:
(a) ∗ is not associative.
(b) V is a vector space over F.
(c) V is not closed under vector addition.
10. Which of the following statements is always TRUE for any real vector space V :
(a) Scalar multiplication of 0 and a vector v is scalar 0.
(b) Scalar multiplication of α ∈ R and the 0 vector is scalar 0.
(c) If α · v = v for some α ∈ R and 0 ̸= v ∈ V then α = 1.
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(d) For any vector u, there is always a vector v such that u + v ̸= 0.
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