Question

Find the inverse of the following matrix.

`[(0,1,2),(1,2,3),(3,1,1)]`

Answer 1

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

∴ |A| = `[(0,1,2),(1,2,3),(3,1,1)]`

= 0(2 − 3) − 1(1 − 9) + 2(1 − 6)
= 0 + 8 − 10
= −2 `≠` 0

∴ A⁻¹ exists.

consider AA⁻¹ = I

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

By R₁ ↔ R₂, we get,

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

By R₃ – 3R₁, we get,

`[(1,2,3),(0,1,2),(0,-5,-8)]`A⁻¹ = `[(0,1,0),(1,0,0),(0,-3,1)]`

By R₁ – 2R₂ and R₃ + 5R₂, we get,

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

By `(1/2)`R₃, we get,

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

By R₁ + R₃ and R₂ − 2R₃, we get,

`[(1,0,0),(0,1,0),(0,0,1)]`A⁻¹ = `[(1//2,-1//2,1//2),(-4,3,-1),(5//2,-3//2,1//2)]`

∴ A⁻¹ = `1/2[(1,-1,1),(-8,6,-2),(5,-3,1)]`

[Note: Answer in the textbook is incorrect.]