Question on System of Linear Equation with Answers

**Question on System of Linear Equation with Answers**

**Q1. The row space of a 20×50 matrix A has dimension 13. What is the dimension of the space of solution A _{x} = 0 ?**

(a) 7

(b) 13

(c) 33

(d) 37

**Q2. Let A be an m×n matrix of rank n with real entries. Choose the correct statement.**

(a) Ax = b has a solution for any b.

(b) Ax = 0 does not have a solution.

(c) If Ax = b has a solution, then it is unique

(d) y’A = 0 for some nonzero y, where y’ denotes the transpose of the vector y.

**Q3. Let A be a 3×4 and b a 3×1 matrix with integer entries. Suppose that the system A _{x} = b has a complex solution then**

(a) A_{x} = b has an integer solution

(b) A_{x} = b has a rational solution

(c) The set of real solution to A_{x} = 0 has a basis consisting of rational solution

(d) If b ≠ 0 then A has positive rank.

**Q4. Let A be a 4×7 real matrix and B be a 7×4 real matrix such that AB = I _{4} where I_{4} is thr 4×4 identity matrix which of the following is/are always true ?**

(a) Rank (A) = 4

(b) Rank (B) = 7

(c) Nullity (B) = 0

(d) BA = I_{7} , where I_{7} is the 7×7 identity matrix

**Q5. Consider a homogeneous system of linear equation A _{x} = 0 where A is an m×n real matrix and n > m. Then which of the following statements are always true?**

(a) A_{x} = 0 has a solution

(b) A_{x} = 0 has no nonzero solution

(c) A_{x} = 0 has a nonzero solution

(d) Dimension of the space of all solutions is at least n – m

**Q6. Let A be a 5×4 matrix with real entries such that A _{x} = 0 if and only if x = 0 where x is a 4×1 vector and 0 is a null vector. Then, the rank of A is**

(a) 4

(b) 5

(c) 2

(d) 1

**Q7. The system of equations**

**x + y + z = 1****2x + 3y – z = 5****x + 2y – kz = 4 , where k ∈ R**

**has an infinite number of solutions for**

(a) k = 0

(b) k = 1

(c) k = 2

(d) k = 3

**Q8. Let A be a 5×4 matrix with real entries such that the space of all solutions of the linear system AX ^{t} = [1,2,3,4, 5]^{t} is given by {[1+2s, 2+3s, 3+4s, 4 +5s]^{t} : s ∈R} . (Here M^{t} denotes the transpose of a matrix M ). Then the rank of A is equal to**

(a) 4

(b) 3

(c) 2

(d) 1

**Q9. Let D be a non zero nxn real matrix with n ≥ 2. Which of the following implications is valid?**

(a) det (D) = 0 implies ranks (D) = 0

(b) det (D) =1 implies ranks (D) ≠ 1

(c) det (D) = 1 implies ranks (D) = 0

(d) det (D) = n implies ranks (D) ≠ 1

**Q10. Consider the system of m linear equations in n unknowns given by A _{x} = b, where A = (a_{ij}) is a real m×n matrix, x and b are n×1 column vectors. Then**

(a) There is at least one solution

(b) There is at least one solution if b is the zera vector

(c) If m = n and if the rank of A is n, then there is a unique sołution

(d) If m < n and if the rank of the augmented matrix [A: b] equals the rank of A, then there are infinitely many solutions.

**Q11. The system of simultaneous linear equations** **x + y + z = 0 , x – y – z = 0 Has**

(a) No solution in R^{3}

(b) A unique solution in R^{3}

(c) Infinitely many solutions in R^{3}

(d) More than 2 but finitely many solution in R^{3}

**Q12. Let A and B be upper and lower triangular matrices given by**

**Then**

(a) A is invertible and B is singular

(b) A is singular and B is invertible

(c) Both A and B are invertible

(d) Neither A and B is invertible.

**Q13. A homogenous system of 5 linear equations in 6 variables admits**

(a) No solution in R^{3}

(b) A unique solution in R^{3}

(c) Infinitely many solution in R^{3}

(d) Finite, but more than 2 solutions in R^{3}

**Q14. Let A be an m×n a matrix with rank m and B be an p×m a matrix with rank p. What will be the rank of BA? (p < m < n)**

(a) m

(b) p

(c) n

(d) p+m

**Q15. Let A be an n×n matrix and b = (b _{1},b_{2},…,b_{n})^{t} be a fixed vector. Consider a system of n-linear equation A_{x} = b, where**

**x = (x**

_{1},x_{2},…,x_{n}). Consider the following statements:**A. If rank A = n, the system has a unique solution****B. If rank A < n, the system has infinitely many solutions****C. If b = 0, the system has at least one solution**

**Which of the following is correct?**

(a) A and B are true

(b) A and C are true

(c) Only A is true

(d) Only B is true

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*Answer Key*

*Answer Key*

01. | (d) | 06. | (a) | 11. | (c) |

02. | (c) | 07. | (c) | 12. | (a) |

03. | (b), (c), (d) | 08. | (b) | 13. | (c) |

04. | (a), (c) | 09. | (b), (c), (d) | 14. | (b) |

05. | (a), (c), (d) | 10. | (b), (c), (d) | 15. | (b) |