# IIT JAM Mathematics Test Series 2023 : Linear Algebra – System of Linear Equation

Practice IIT JAM Mathematics Test Series for Free only On www.examflame.com and take your preparation to the another level. In the Series of Tests for IIT JAM 2023 this is the test of topic System of Linear Equation of the Chapter Linear Algebra. These Questions are prepared as per the Latest Syllabus of IIT JAM Mathematics 2023 . Practicing mock tests/Test series, you will get an idea about how and which type of Question will ask in the Examination. It also boost your confidence level. Also Solve IIT JAM Mathematics Previous Year Question Paper. And Don’t Forget to Shore with Your Friends.

## IIT JAM Mathematics Test Series 2023 : Linear Algebra – System of Linear Equation

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

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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