Question

Find the inverse of the following matrix.

`[(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, we get,

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

By R₁ → R₁ − R₂, We get,

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

By R₂ → R₂ − 5R₁, We get,

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

By R₂ → R₂ − 5R₃, We get,

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

By R₁→R₁ + R₂ and R₃ →R₃ − R₂, we get,

`[(1,0,-3),(0,1,0),(0,0,3)]"A"^(−1) = [(-12,5,-5),(- 15,6,-5),(15,-6,6)]`

By R₃ →`(1/3)`R₃, We got,

`[(1,0,-3),(0,1,0),(0,0,1)]"A"^(−1) = [(-12,5,-5),(- 15,6,-5),(5,-2,2)]`

By R₁ →R₁ + 3R₃, we get,

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

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