Question

Find the inverse (if it exists) of the following:

`[(2, 3, 1),(3, 4, 1),(3, 7, 2)]`

Answer 1

A = `[(2, 3, 1),(3, 4, 1),(3, 7, 2)]`

|A| = 2(8 – 7) – 3(6 – 3) + 1(21 – 12)

= 2 – 9 + 9

= 2 ≠ 0. A^-1 exists.

adj A = `[(+|(4, 1),(7, 2)|, -|(3, 1),(3, 2)|, +|(3, 4),(3, 7)|),(-|(3, 1),(7, 2)|, +|(2, 1),(3, 2)|, -|(2, 3),(3, 7)|),(+|(3, 1),(4, 1)|, -|(2, 1),(3, 1)|, +|(2, 3),(3, 4)|)]^"T"`

= `[(+(8 - 7), -(6 - 3), +(21 - 12)),(-(6 - 7), +(4 - 3), -(14 - 9)),(+(3 - 4), - (2 - 3), +(8 - 9))]^"T"`

= `[(1, -3, 9),(1, 1, -5),(-1, 1, -1)]^"T"`

∴  adj A = `[(1,  1, -1),(-3, 1, 1),(9, -5, -1)]`

A^-1 = `1/|"A"|`

adj A = `1/2 [(1, 1, -1),(-3, 1, 1),(9, -5, -1)]`