Question

Find the matrix B if A = `[(4, 1),(2, 3)]` and A² = A + 2B

Answer 1

A = `[(4, 1),(2, 3)]`

Let  B = `[(a, b),(c, d)]`

A² = A x A = `[(4, 1),(2, 3)][(4, 1),(2, 3)]`

= `[(16 + 2, 4 + 3),(8 + 6, 2 + 9)]`

= `[(18, 7),(14, 11)]`

A + 2B = `[(4, 1),(2, 3)] + 2[(a, b),(c, d)]`

= `[(4, 1),(2, 3)] + [(2a, 2b),(2c, 3 + 2d)]`

= `[(4 + 2a, 1 + 2b),(2 + 2c, 3 + 2d)]`
∵ A2 = A + 2B

∴ `[(18, 7),(14, 11)] = [(4 + 2a, 1 + 2b),(2 + 2c, 3 + 2d)]`
Comparing the corresponding elements
4 + 2a = 18
⇒ 2a = 18 – 4 = 14
∴ a = 7
1 + 2b = 7
⇒ 2b = 7 – 1 = 6
∴ b = 3
2 + 2c = 14
⇒ 2c = 14 – 2 = 12
∴ c = 6
3 + 2d = 11
⇒  2d = 11 – 3 = 8
∴ d = 4
Hence a = 7, b = 3, c = 6, d = 4
∴ B = `[(7, 3),(6, 4)]`.