Question

If B = `[(-4, 2),(5, -1)] and "C" = [(17, -1),(47, -13)]` find the matrix A such that AB = C

Answer 1

B = `[(-4, 2),(5, -1)]`

C = `[(17, -1),(47, -13)]` 
and AB = C

Let A = `[(a, b),(c, d)]`

Then AB = `[(a, b),(c, d)] xx [(-4, 2),(5, -1)]`

= `[(-4a + 5b, 2a - b),(-4c + 5d, 2c - d)]`
∵ AB = C

∴ `[(-4a + 5b, 2a - b),(-4c + 5d, 2c - d)] = [(17, -1),(47, 13)]`
Comparing corresponding elements, we get
∵ –4a + 5b = 17                         ....(i)
2a – b = –1                                 ....(ii)
–4c + 5d = 47                            ....(iii)
2c –d = –13                                ....(iv)
Multiplying (i) by 1 and (ii) by 2
⇒ –4a + 5b = 17
      4a – 2b = –2
Adding
3b = 15
⇒ b = `(15)/(3)` = 5
2a – b = –1
⇒ 2a – 5 = –1
⇒ 2a = –1 + 5 = 4
⇒ a = `(4)/(2) + 2`
∴ a = 2, b = 5
Again multiplying (iii) by 1 and (iv) by 2,
–4c + 5d = 47
4c – 2d = –26
Adding
3d = 21
⇒ d = `(21)/(3)` = 7
and
2c – d = –13
⇒ 2c – 7 = –13
⇒ 2c = –13 + 7 = –6
⇒ c = `(-6)/(2)` = –3
∴ c = –3, d = 7
Now matrix A = `[(2, 5),(-3, 7)]`.