Question

Find the inverse of the following matrices by transformation method:

`[(2, 0, −1),(5, 1, 0),(0, 1, 3)]`

Answer 1

Let A = `[(2, 0, −1),(5, 1, 0),(0, 1, 3)]`

∴ |A| = `|(2, 0, -1),(5, 1, 0),(0, 1, 3)|`

= 2(3 – 0) – 0(15 – 0) – 1(5 – 0)
= 6 – 0 – 5
= 1 ≠ 0
∴ A^–1 exists.
Consider AA^–1 = I

∴ `[(2, 0, –1),(5, 1, 0),(0, 1, 3)] "A"^-1 = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`

By R₁→3R₁,

`[(6, 0, –3),(5, 1, 0),(0, 1, 3)] "A"^-1 = [(3, 0, 0),(0, 1, 0),(0, 0, 1)]`

By R₁ → R₁ − R₂

`[(1, –1, –3),(5, 1, 0),(0, 1, 3)] "A"^-1 = [(3, –1, 0),(0, 1, 0),(0, 0, 1)]`

By R₂ → R₂ − 5R₁

`[(1, –1, –3),(0, 6, 15),(0, 1, 3)] "A"^-1 = [(3, –1, 0),(–15, 6, 0),(0, 0, 1)]`

By R₂↔R₃

`[(1, –1, –3),(0, 1, 3),(0, 6, 16)] "A"^-1 = [(3, –1, 0),(0, 0, 1),(–15, 6, 0)]`

By R₁→R₁ + R₂ and R₃→R₃ − 6R₂

`[(1, 0, 0),(0, 1, 3),(0, 0, –3)] "A"^-1 = [(3, –1, 1),(0, 0, 1),(–15, 6, –6)]`

By R₃ → `(- 1)/3 "R"_3`,

`[(1, 0, 0),(0, 1, 3),(0, 0, 1)] "A"^-1 = [(3, –1, 1),(0, 0, 1),(5, –2, 2)]`

By R₂ →R₂ – 3R₃

`[(1, 0, 0),(0, 1, 0),(0, 0, 1)] "A"^-1 = [(3, –1, 1),(–15, 6, –5),(5, –2, 2)]`

IA^–1 = `[(3, –1, 1),(–15, 6, –5),(5, –2, 2)]`

A^–1 = `[(3, –1, 1),(–15, 6, –5),(5, –2, 2)]`