Question

Find the inverse of the following matrix (if they exist):

`[(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 - 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 ↔ R₂, we get,

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

By R₁ - 2R₂, we get,

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

By R₂ - 2R₁, we get,

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

By R₂ + 3R₃, we get,

`[(1,1,2),(0,1,4),(0,1,3)] "A"^-1 = [(-2,1,0),(5,-2,3),(0,0,1)]`

By R₁ - R₂ and R₃ - R₂, we get,

`[(1,0,-2),(0,1,4),(0,0,-1)] "A"^-1 = [(-7,3,-3),(5,-2,3),(-5,2,-2)]`

By (- 1)R₃, we get,

`[(1,0,-2),(0,1,4),(0,0,1)] "A"^-1 = [(-7,3,-3),(5,-2,3),(5,-2,2)]`

By `"R"_1 + 2"R"_3` and  `"R"_2 - 4"R"_3`, we get,

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

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