Question

A = `[(3, 0),(4, 5)]`, B = `[(6, 3),(8, 5)]`, C = `[(3, 6),(1, 1)]` find the matrix D, such that CD – AB = 0

Answer 1

Given A = `[(3, 0),(4, 5)]`, B = `[(6, 3),(8, 5)]`, C = `[(3, 6),(1, 1)]`

Let D be `[("a", "b"),("c","d")]`

Given CD – AB = 0

AB = CD

`[(3, 0),(4, 5)] [(6, 3),(8, 5)] = [(3, 6),(1, 1)] [("a", "b"),("c", "d")]`

`[(18 + 0, 9 + 0),(24 + 40, 12 + 25)] = [(3"a" + 6"c", 3"b" + 6"d"),("a" + "c", "b" + "d")]`

`[(18, 9),(64, 37)] = [(3"a" + 6"c", 3"b" + 6"d"),("a" + "c", "b" + "d")]`

            3a + 6c = 18
(÷ 3) ⇒   a + 2c = 6   ....(1)
               a + c = 64   ....(2)
           (–)  (–)  (–)     
(1) – (2) ⇒     c = – 58

Substitute the value of c = – 58 in (1)
            a – 116  = 6
                     a  = 6 + 116
                         = 122
           3b + 6d = 9
(÷ 3) ⇒ b + 2d = 3      ....(3)
             b +  d = 37    ....(4)
            (–) (–)  (–)       
(3) – (4) ⇒     d = – 34

Substitute the value of d = – 34 in (4)

b – 34 = 37

b = 37 + 34

= 71

Matrix D = `[(122, 71),(- 58, - 34)]`