Question

Find the inverse of each of the matrices, if it exists.

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

Answer 1

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

We know that A = IA

`[(2, 0, -1),(5, 1, 0),(0, 1, 3)] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]` A, R₁ ⇔ R₂

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

R₁ - 2R₂ ⇒ R₁

=> `[(1, 1, 2),(2, 0, -1),(0, 1, 3)]= [(-2, 1, 0),(1, 0, 0),(0, 0, 1)]`

A, R₂ − 2R₁ ⇒ R₂

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

A; R₂ ⇒ −R₂

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

A; R₂ ⇒ R₂

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

A; R₂ − R₃ ⇒ R₂

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

A; R₁ − R₂ ⇒ R₁

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

A; R₃ − R₂ ⇒ R₃

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

A; R₂ − 2R₃ ⇒ R₂

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