Question

If possible, using elementary row transformations, find the inverse of the following matrices

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

Answer 1

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

Put A = IA

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

R₁ → 3R₁ – R₂

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

R₂ → R₂ – 5R₁ 

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

R₂ → R₂ – 5R₃ 

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

R₃ → R₃ – R₂  

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

R₁ → R₁ + R₂  

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

`"R"_3 -> 1/3 "R"_3`

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

R₁ → R₁ + 3R₃  

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

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