Choose the correct answer

Choose the correct answer

1. If A=(1 2 3), then the rank of AAT is

(a) 0

(b) 2

(c) 3

(d) 1

2. The rank of m × n matrix whose elements are unity is

(a) 0

(b) 1

(c) m

(d) n

3. If   is a transition probability matrix, then at equilibrium A is equal to

(a) 1/4

(b) 1/5

(c) 1/6

(d) 1/8

4. If A =  , then ρ(A) is

(a) 0

(b) 1

(c) 2

(d) n

5. The rank of the matrix  is

 (a) 0

(b) 1

(c) 2

(d) 3

6. The rank of the unit matrix of order n is

(a) n −1

(b) n

(c) n + 1

(d) n2

7. If ρ (A) = r then which of the following is correct?

(a) all the minors of order r which does not vanish

(b) A has at least one minor of order r which does not vanish

(c) A has at least one (r+1) order minor which vanishes

(d) all (r+1) and higher order minors should not vanish

8. If A =  then the rank of AAT is

(a) 0

(b) 1

(c) 2

(d) 3

9. If the rank of the matrix  is 2. Then λ is

(a) 1

(b) 2

(c) 3

(d) only real number

10. The rank of the diagonal matrix 

 (a) 0

(b) 2

(c) 3

(d) 5

11. If T =  is a transition probability matrix, then the value of x is

(a) 0.2

(b) 0.3

(c) 0.4

(d) 0.7

12. Which of the following is not an elementary transformation?

(a) Ri ↔ Rj

(b) Ri → 2 Ri + 2Cj

(c) Ri → 2 Ri − 4Rj

(d) Ci → Ci + 5Cj

13. If ρ(A) = ρ(A, B) then the system is

(a) Consistent and has infinitely many solutions

(b) Consistent and has a unique solution

(c) Consistent

(d) inconsistent

14. If ρ(A) = ρ(A, B) the number of unknowns, then the system is

(a) Consistent and has infinitely many solutions

(b) Consistent and has a unique solution

(c) inconsistent

(d) consistent

15. If ρ ( A) ≠ ρ ( A, B) , then the system is

(a) Consistent and has infinitely many solutions

(b) Consistent and has a unique solution

(c) inconsistent

(d) consistent

16. In a transition probability matrix, all the entries are greater than or equal to

(a) 2

(b) 1

(c) 0

(d) 3

17. If the number of variables in a non- homogeneous system AX = B is n, then the system possesses a unique solution only when

(a) ρ ( A ) = ρ ( A, B ) > n

(b) ρ ( A ) = ρ ( A, B ) = n

(c) ρ ( A ) = ρ ( A, B ) < n

(d) none of these

18. The system of equations 4x + 6 y = 5, 6x + 9 y = 7 has

(a) a unique solution

(b) no solution

(c) infinitely many solutions

(d) none of these

19. For the system of equations x + 2 y + 3z = 1, 2x + y + 3z = 2 5x + 5y + 9z = 4

(a) there is only one solution

(b) there exists infinitely many solutions

(c) there is no solution

(d) None of these

20. If |A| ≠ 0, then A is

(a) non- singular matrix

(b) singular matrix

(c) zero matrix

(d) none of these

21. The system of linear equations x + y + z = 2, 2x + y − z = 3, 3x + 2 y + k = 4 has unique solution, if k is not equal to

(a) 4

(b) 0

(c) –4

(d) 1

22. Cramer’s rule is applicable only to get an unique solution when

(a) Δz ≠ 0

(b) Δx ≠ 0

(c) Δ ≠ 0

(d) Δy ≠ 0

23. If 

 then (x y) is

Ans: (d)

24. |An ×n| = 3 |adjA| = 243 then the value n is

(a) 4

(b) 5

(c) 6

(d) 7

25. Rank of a null matrix is

(a) 0

(b) –1

(c) ∞

(d) 1