Question

Find the matrix A such that `[(2, -1),(1, 0),(-3, 4)]"A"^"T" = [(-1, -8, -10),(1, 2, -5),(9, 22, 15)]`

Answer 1

`[(2, -1),(1, 0),(-3, 4)]"A"^"T" = [(-1, -8, -10),(1, 2, -5),(9, 22, 15)]`

Let A^T = `[("a", "b","c"),("d", "e", "f")]` then

`[(2, -1),(1, 0),(-3, 4)] [("a", "b","c"),("d", "e", "f")] = [(-1, -8, -10),(1, 2, -5),(9, 22, 15)]`

`[(2"a" - "d", 2"b" - "e", 2"c" - "f"),("a" + 0, "b" + 0, "c" + 0),(-3"a" + "d", -3"b" + 4"e", -3"c" + 4"f")] = [(-1, -8, -10),(1, 2, -5),(9, 22, 15)]`

Equating like entries

2a – d = –1, 2b – e = – 8, 2c – f = – 10

a = 1, b = 2, c = -5

2a – d = – 1

⇒ 2 × 1 – d = – 1

⇒ 2 + 1 = d

⇒ d = 3

2b – e = – 8

⇒ 2 × 2 – e

= – 8

⇒ 4 + 8 = e

⇒ e = 12

2c – f = – 10

⇒ 2 × – 5 – f = – 10

⇒ – 10 – f = – 10

⇒ f = 0

∴ A^T = `[(1, 2, -5),(3, 12, 0)]`

A = `[(1, 3),(2, 12),(-5, 0)]`