Choose the Correct Answers

Choose the Correct or the most suitable answer from the given four alternatives: 

1. If | adj(adj A) |=| A |9 , then the order of the square matrix A is

(1) 3

(2) 4

(3) 2

(4) 5

2. If A is a 3× 3 non-singular matrix such that AAT = AT A and B = A−1 AT , then BBT =

(1) A

(2) B

(3) I3

(4) BT

3. If A =  , B = adj A and C = 3A , then | adj B | / |C| =

(1) 1/3

(2) 1/9

(3) 1/4

(4) 1

4. if , then A = 

Ans: (3)

5. if  , then 9I2 – A =

(1) A-1

(2) A-1/2

(3) 3A-1

(4) 2A-1

6. if  then | adj ( AB) |=

(1) -40

(2) -80

(3) -60

(4) -20

7. If P=  is the adjoint of 3× 3 matrix A and | A |= 4 , then x is

(1) 15

(2) 12

(3) 14

(4) 11

8.If A= , and A−1 =  then the value of a23 is

(1) 0

(2) -2

(3) -3

(4) -1

9. If A, B and C are invertible matrices of some order, then which one of the following is not true?

(1) adj A =| A | A-1

(2) adj( AB) = (adj A)(adj B)

(3) det A-1 = (det A)-1

(4) ( ABC)-1 = C-1B-1A-1

10. If ( AB)−1 = , and A-1 = , then B−1 = 

Ans: (1)

11. If AT A−1 is symmetric, then A2 =

(1) A-1

(2) (AT)2

(3) AT

(4) ( A-1 )2

12. If A is a non-singular matrix such that A−1  = , then ( AT )−1 =

Ans: (4)

13.if A= ,  and AT = A−1 , then the value of x is

(1) -4/5

(2) -3/5

(3) 3/5

(4) 4/5

Ans: (1)

14.if A= , and AB = I2 , then B=

Ans: (2)

15. if A= , and A(adj A) = , then k=

(1) 0

(2) sinθ

(3) cosθ

(4) 1

Ans: (4)

16. If A= , be such that λ A−1 = A , then λ is

(1) 17

(2) 14

(3) 19

(4) 21

17.If adj A= , adj B= , then adj ( AB) is

Ans: (2)

18. The rank of the matrix  is

(1) 1

(2) 2

(3) 4

(4) 3

Ans: (1)

19. If xa yb = em , xc yd = en ,   then the values of x and y are respectively,

(1) e(Δ2 / Δ1 ) , e(Δ3 / Δ1 )

(2) log(Δ1/ Δ3 ), log(Δ2/Δ3)

(3) log(Δ2/ Δ1 ), log(Δ3/Δ1)

(4) ) e(Δ1 / Δ3 ) ,e(Δ2/Δ3 )

20. Which of the following is/are correct?

(i) Adjoint of a symmetric matrix is also a symmetric matrix.

(ii) Adjoint of a diagonal matrix is also a diagonal matrix.

(iii) If A is a square matrix of order n and λ is a scalar, then adj( λ A) =  λn adj( A) .

(iv) A(adjA) = (adjA) A =| A | I

(1) Only (i) 

(2) (ii) and (iii) 

(3) (iii) and (iv) 

(4) (i), (ii) and (iv) 

21. If ρ ( A) = ρ ([ A | B]) , then the system AX = B of linear equations is

(1) consistent and has a unique solution

(2) consistent

(3) consistent and has infinitely many solution

(4) inconsistent

22. If 0 ≤ θ ≤ π and the system of equations x + (sinθ ) y − (cosθ )z = 0, (cosθ )x − y + z = 0, (sinθ )x + y − z = 0 has a non-trivial solution then θ is

(1) 2π/3

(2) 3π/4

(3) 5π/6

(4) π/4

23. The augmented matrix of a system of linear equations is  . The system has infinitely many solutions if

(1) λ = 7, μ ≠ -5

(2) λ = -7, μ = 5

(3) λ ≠ 7, μ ≠ -5

(4) λ = 7, μ = -5

24. let A = , and 4B = , If B is the inverse of A , then the value of x is

(1) 2

(2) 4

(3) 3

(4) 1

25.If A= , then adj(adj A) is

Ans: (1)

Choose the Correct or the most suitable answer from the given four alternatives: 

1. If | adj(adj A) |=| A |9 , then the order of the square matrix A is

(1) 3

(2) 4

(3) 2

(4) 5

2. If A is a 3× 3 non-singular matrix such that AAT = AT A and B = A−1 AT , then BBT =

(1) A

(2) B

(3) I3

(4) BT

3. If A =  , B = adj A and C = 3A , then | adj B | / |C| =

(1) 1/3

(2) 1/9

(3) 1/4

(4) 1

4. if , then A = 

Ans: (3)

5. if  , then 9I2 – A =

(1) A-1

(2) A-1/2

(3) 3A-1

(4) 2A-1

6. if  then | adj ( AB) |=

(1) -40

(2) -80

(3) -60

(4) -20

7. If P=  is the adjoint of 3× 3 matrix A and | A |= 4 , then x is

(1) 15

(2) 12

(3) 14

(4) 11

8.If A= , and A−1 =  then the value of a23 is

(1) 0

(2) -2

(3) -3

(4) -1

9. If A, B and C are invertible matrices of some order, then which one of the following is not true?

(1) adj A =| A | A-1

(2) adj( AB) = (adj A)(adj B)

(3) det A-1 = (det A)-1

(4) ( ABC)-1 = C-1B-1A-1

10. If ( AB)−1 = , and A-1 = , then B−1 = 

Ans: (1)

11. If AT A−1 is symmetric, then A2 =

(1) A-1

(2) (AT)2

(3) AT

(4) ( A-1 )2

12. If A is a non-singular matrix such that A−1  = , then ( AT )−1 =

Ans: (4)

13.if A= ,  and AT = A−1 , then the value of x is

(1) -4/5

(2) -3/5

(3) 3/5

(4) 4/5

Ans: (1)

14.if A= , and AB = I2 , then B=

Ans: (2)

15. if A= , and A(adj A) = , then k=

(1) 0

(2) sinθ

(3) cosθ

(4) 1

Ans: (4)

16. If A= , be such that λ A−1 = A , then λ is

(1) 17

(2) 14

(3) 19

(4) 21

17.If adj A= , adj B= , then adj ( AB) is

Ans: (2)

18. The rank of the matrix  is

(1) 1

(2) 2

(3) 4

(4) 3

Ans: (1)

19. If xa yb = em , xc yd = en ,   then the values of x and y are respectively,

(1) e(Δ2 / Δ1 ) , e(Δ3 / Δ1 )

(2) log(Δ1/ Δ3 ), log(Δ2/Δ3)

(3) log(Δ2/ Δ1 ), log(Δ3/Δ1)

(4) ) e(Δ1 / Δ3 ) ,e(Δ2/Δ3 )

20. Which of the following is/are correct?

(i) Adjoint of a symmetric matrix is also a symmetric matrix.

(ii) Adjoint of a diagonal matrix is also a diagonal matrix.

(iii) If A is a square matrix of order n and λ is a scalar, then adj( λ A) =  λn adj( A) .

(iv) A(adjA) = (adjA) A =| A | I

(1) Only (i) 

(2) (ii) and (iii) 

(3) (iii) and (iv) 

(4) (i), (ii) and (iv) 

21. If ρ ( A) = ρ ([ A | B]) , then the system AX = B of linear equations is

(1) consistent and has a unique solution

(2) consistent

(3) consistent and has infinitely many solution

(4) inconsistent

22. If 0 ≤ θ ≤ π and the system of equations x + (sinθ ) y − (cosθ )z = 0, (cosθ )x − y + z = 0, (sinθ )x + y − z = 0 has a non-trivial solution then θ is

(1) 2π/3

(2) 3π/4

(3) 5π/6

(4) π/4

23. The augmented matrix of a system of linear equations is  . The system has infinitely many solutions if

(1) λ = 7, μ ≠ -5

(2) λ = -7, μ = 5

(3) λ ≠ 7, μ ≠ -5

(4) λ = 7, μ = -5

24. let A = , and 4B = , If B is the inverse of A , then the value of x is

(1) 2

(2) 4

(3) 3

(4) 1

25.If A= , then adj(adj A) is

Ans: (1)