PRACTICE ON MATRICES AND DETERMINANTS WORKSHEET

Problem 1 :

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

(1)  3     (2)  4     (3)  2    (4)  5

Solution

Problem 2 :

If A is a 3 × 3 non - singular matrix such that AA^T = A^T A and B = A^-1 A^T, then BB^T = 

(1)  A     (2)  B     (3)  I₃    (4)  B^T

Solution

Problem 3 :

(1)  1/3      (2)  1/9    (3)  1/4      (4)  1

Solution

Problem 4 :

Solution

Problem 5 :

(1) A^-1       (2) A^-1/2      (3) 3A^-1    (4)  2A^-1

So, option 4) is correct.

Solution

Problem 6 :

1) -40     2) -80     3) -60      4) -20

Solution

Problem 1 :

(1) 15     (2) 12     (3) 14     (4) 11

Solution

Problem 2 :

(1) 0     (2) -2     (3) -3     (4) -1

Solution

Problem 3 :

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

Solution

Problem 4 :

Solution

Problem 5 :

If A^T A^-1 is symmetric, then A² = 

(1) A^-1     (2) (A^T)²     (3) A^T     (4) (A^-1)²

Solution

Problem 6 :

Solution

Problem 1 :

Solution

Problem 2 :

Solution

Problem 3 :

Solution

Problem 4 :

Solution

Problem 5 :

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(λA) = λⁿadj(A)

(iv) A(adj A) = (adj A)A = |A|I

(i)  Only (i)  (2) (ii) and (iii)  (3)  (iii) and (iv)  (4)  (i), (ii) and (iv)

Solution

Problem 6 :

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

Solution

Problem 7 :

If 0 ≤ θ ≤ π and the system of equations x + (sin θ)y – (cos θ)z = 0, (cos θ)x – y + z = 0, (sin θ)x + y – z = 0 has a non – trivial solution then θ is

Solution