Question

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

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

Answer 1

Here, A = `[(2, -1, 3),(-5, 3, 1),(-3, 2, 3)]` for elementary row transformation

We put A = IA

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

R₂ → R₂ + R₁

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

R₃ → R₃ – R₂ 

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

R₁ → R₁ + R₂ 

`[(-1, 1, 7),(0, -1, -17),(0, 0, -1)] = [(2, 1, 0),(-5, -2, 0),(-1, -1, 1)]"A"`

R₁ → R₁ + R₂ and R₃ → –1 . R₃ 

`[(-1, 0, -10),(0, -1, -17),(0, 0, -1)] = [(-3, -1, 0),(-5, -2, 0),(-1, -1, 1)]"A"`

R₁ → R₁ + 10R₃ and R₂ → R₂ + 17R₃

`[(-1, 0, 0),(0, -1, 0),(0, 0, 1)] = [(7, 9, -10),(12, 15, -17),(1, 1, -1)]"A"`

R₁ → – 1.R₁ and R₂ → – 1.R₂

`[(1, 0, 0),(0, 1, 0),(0, 0, 1)] = [(-7, 9-, 10),(-12, -15, 17),(1, 1, -1)]"A"`

Hence, A^–1 = `[(-7, 9-, 10),(-12, -15, 17),(1, 1, -1)]`