Question 1

Question 1: Given A = , I = if A – λI is a singular matrix, then ___________ (a) λ = 0 (b) λ2 – 3λ – 4 = 0 (c) λ2 + 3λ – 4 = 0 (d) λ2 – 3λ – 6 = 0

Accepted Answer

(b) λ2 – 3λ – 4 = 0 Hint:

Question 2

Question 2: Consider the matrices A = , B = , C = . Out of the given matrix products, ___________ (i) (AB)TC (ii) CTC(AB)T (iii) CTAB (iv) ATABBTC (a) Exactly one is defined (b) Exactly two ar

Accepted Answer

(c) Exactly three are defined Hint: A is of order 3 × 3, B is of order 3 × 2 and C is of order 3 × 1. (AB)TC is of order 2 × 1. CTC and (AB)T are of different orders. CTC (AB)T is not defined. CTAB is of order 1 × 2. ATABBTC is of order 3 × 1.

Question 3

Question 3: If A and B are square matrices of equal order, then which one is correct among the following? (a) A + B = B + A (b) A + B = A – B (c) A – B = B – A (d) AB = BA

Accepted Answer

(a) A + B = B + A Hint: Matrix addition is commutative. ∴ A + B = B + A

Question 4

Question 4: If A = is a matrix satisfying the equation AAT = 9I, where I is the identity matrix of order 3, then the ordered pair (a, b) is equal to ___________ (a) (2, -1) (b) (-2, 1) (c) (2,

Accepted Answer

(d) (-2, -1)

Question 5

Question 5: If A = and |A3| = 125, then α = ___________ (a) ±3 (b) ±2 (c) ±5 (d) 0

Accepted Answer

(a) ±3 Hint: |A3| = 125 |A|3 = 53 …….[∵ |An| = |A|n, n ∈ N] ∴ |A| = 5 α2 – 4 = 5 α2 = 9 ∴ α = ± 3

Question 6

Question 6: If , then ___________ (a) x = 1, y = -2 (b) x = -1, y = 2 (c) x = 1, y = 2 (d) x = -1, y = -2

Accepted Answer

(c) x = 1, y = 2

Question 7

Question 7: If A + B = and A – B = , then the value of A is ___________ (a) (b) (c) (d)

Accepted Answer

(b)

Question 8

Question 8: If , then ___________ (a) x = 3, y = 7, z = 1, w = 14 (b) x = 3, y = -5, z = -1, w = -4 (c) x = 3, y = 6, z = 2, w = 7 (d) x = -3, y = -7, z = -1, w = -14

Accepted Answer

(a) x = 3, y = 7, z = 1, w = 14

Question 9

Question 9: For suitable matrices A, B, the false statement is ___________ (a) (AB)T = ATBT (B) (AT)T = A (C) (A – B)T = AT – BT (D) (A + B)T = AT + BT

Accepted Answer

(a) (AB)T = ATBT Hint: (AB)T = BTAT

Question 10

Question 10: If A = and f(x) = 2x2 – 3x, then f(A) = ___________ (a) (b) (c) (d)

Accepted Answer

(c) Hint:

Question 11

Question 1: If A = diag[2, -3, -5], B = diag[4, -6, -3] and C = diag [-3, 4, 1], then find i. B + C – A ii. 2A + B – 5C.

Accepted Answer

Step-by-step solution provided.

Question 12

Question 2: If f(α) = A = , find i. f(-α) ii. f(-α) + f(α)

Accepted Answer

Step-by-step solution provided.

Question 13

Question 3: Find matrices A and B, where (i) 2A – B = and A + 3B = (ii) 3A – B = and A + 5B =

Accepted Answer

i. Given equations are 2A – B = …….(i)

Question 14

Question 4: If A = , B = , verify i. (A + 2BT)T = AT + 2B ii. (3A – 5BT)T = 3AT – 5B.

Accepted Answer

Step-by-step solution provided.

Question 15

Question 5: If A = and A + AT = I, where I is a unit matrix of order 2 × 2, then find the value of α.

Accepted Answer

Step-by-step solution provided.

Question 16

Question 6: If A = and B = , show that AB is singular.

Accepted Answer

Step-by-step solution provided.

Question 17

Question 7: If A = , B = , show that AB and BA are both singular matrices.

Accepted Answer

Step-by-step solution provided.

Question 18

Question 8: If A = , B = , show that BA = 6I.

Accepted Answer

Step-by-step solution provided.

Question 19

Question 9: If A = , B = , verify that |AB| = |A|.|B|.

Accepted Answer

Step-by-step solution provided.

Question 20

Question 10: If Aα = , show that Aα. Aβ = Aα+β

Accepted Answer

Step-by-step solution provided.

Question 21

Question 11: If A = , B = , where ω is a complex cube root of unity, then show that AB + BA + A – 2B is a null matrix.

Accepted Answer

ω is the complex cube root of unity. ω3 = 1 ω3 – 1 = 0 (ω – 1) (ω2 + ω + 1) = 0 ω = 1 or ω2 + ω + 1 = 0 But, ω is a complex number. 1 + ω + ω2 = 0 …….(i) AB + BA + A – 2B which is a null matrix.

Question 22

Question 12: If A = , show that A2 = A.

Accepted Answer

Step-by-step solution provided.

Question 23

Question 13: If A = , show that A2 = I.

Accepted Answer

Step-by-step solution provided.

Question 24

Question 14: If A = , show that A2 – 5A – 14I = 0.

Accepted Answer

Step-by-step solution provided.

Question 25

Question 15: If A = , show that A – 4A + 3I = 0.

Accepted Answer

Step-by-step solution provided.

Question 26

Question 16: If A = , B = and (A + B)(A – B) = A2 – B2, find x and y.

Accepted Answer

(A + B)(A – B) = A2 – B2 A2 – AB + BA – B2 = A2 – B2 -AB + BA = 0 AB = BA By equality of matrices, we get 2 – 4x = -3x ∴ x = 2 and 2y = 2x y = x ∴ y = 2 ∴ x = 2, y = 2

Question 27

Question 17: If A = and B = , show that (A + B)(A – B) ≠ A2 – B2.

Accepted Answer

We have to prove that (A + B). (A – B) ≠ A2 – B2 i.e., to prove that A(A – B) + B(A – B) ≠ A2 – B2 i.e., to prove that A2 – AB + BA – B2 ≠ A2 – B2 i.e., to prove that AB ≠ BA. ∴ AB ≠ BA

Question 28

Question 18: If A = , find A3.

Accepted Answer

∴ A2 = I Multiplying throughout by A, we get A3 = A. I ∴ A3 = A

Question 29

Question 19: Find x, y if,

Accepted Answer

Step-by-step solution provided.

Question 30

Question 20: Find x, y, z if

Accepted Answer

Step-by-step solution provided.

Chapter 4 Determinants and Matrices Miscellaneous Exercise 4(B) Solutions