Question on System of Linear Equation with Answers

Question on System of Linear Equation with Answers

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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 Ax = 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 Ax = b has a complex solution then

(a) Ax = b has an integer solution
(b) Ax = b has a rational solution
(c) The set of real solution to Ax = 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 = I4 where I4 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 = I7 , where I7 is the 7×7 identity matrix


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

(a) Ax = 0 has a solution
(b) Ax = 0 has no nonzero solution
(c) Ax = 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 Ax = 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 AXt = [1,2,3,4, 5]t is given by {[1+2s, 2+3s, 3+4s, 4 +5s]t : s ∈R} . (Here Mt 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 Ax = b, where A = (aij) 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 R3
(b) A unique solution in R3
(c) Infinitely many solutions in R3
(d) More than 2 but finitely many solution in R3


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

IMG 20220816 121901 950 compress15


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 R3
(b) A unique solution in R3
(c) Infinitely many solution in R3
(d) Finite, but more than 2 solutions in R3


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 = (b1,b2,…,bn)t be a fixed vector. Consider a system of n-linear equation Ax = b, where x = (x1,x2,…,xn). 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

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)

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