Q5. Verify (AB)′ = B′A′ for the column A with entries 1, −4, 3 and the row B = [−1, 2, 1] (NCERT Exercise 3.3, Q5 (i), page 67).
AB has rows −1, 2, 1; 4, −8, −4; −3, 6, 3, so (AB)′ has rows −1, 4, −3; 2, −8, 6; 1, −4, 3. B′A′ is the column −1, 2, 1 times the row [1, −4, 3], with the same rows. They agree.
Q6. A has rows sin α, cos α and −cos α, sin α. Verify that A′A = I (NCERT Exercise 3.3, Q6 (ii), page 67).
A′ has rows sin α, −cos α and cos α, sin α. The (1, 1) element of A′A is sin2 α + cos2 α = 1, the (1, 2) element is sin α cos α − cos α sin α = 0, the (2, 1) element is 0 and the (2, 2) element is cos2 α + sin2 α = 1. So A′A = I.
Q7. For A with rows 3, √3, 2 and 4, 2, 0, and B with rows 2, −1, 2 and 1, 2, 4, verify that (A + B)′ = A′ + B′ and that (kB)′ = kB′ (NCERT Example 20, pages 61–62).
A + B has rows 5, √3 − 1, 4 and 5, 4, 4, so (A + B)′ has rows 5, 5; √3 − 1, 4; 4, 4, which is A′ + B′. kB has rows 2k, −k, 2k and k, 2k, 4k, and both (kB)′ and kB′ have rows 2k, k; −k, 2k; 2k, 4k.
Q8. A is 3 × 4 and B is 4 × 2. Which of AB, BA, A′B′, B′A′ and (AB)′ are defined? Give their orders.
AB is (3 × 4)(4 × 2) = 3 × 2. BA would be (4 × 2)(3 × 4): not defined. A′B′ would be (4 × 3)(2 × 4): not defined. B′A′ is (2 × 4)(4 × 3) = 2 × 3, and it equals (AB)′, which is also 2 × 3.
Q9. Show that (ABC)′ = C′B′A′ for any three matrices whose product ABC is defined.
Group the product as (AB)C. Then ((AB)C)′ = C′(AB)′ = C′(B′A′) = C′B′A′, using the product rule twice and the associative law.