Question

If A⁻¹ = `[(3,-1,1),(-15,6,-5),(5,-2,2)]` and B = `[(1,2,-2),(-1,3,0),(0,-2,1)]`, find (AB)⁻¹.

Answer 1

We know that, (AB)⁻¹ = B⁻¹A⁻¹

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

∴ |B| = 1 × 3 − 2 × (−1) − 2(2)

= 3 + 2 − 4

= 5 − 4

= 1

Now, A₁₁ = 3, A₁₂ = 1, A₁₃ = 2

A₂₁ = 2, A₂₂ = 1, A₂₃ = 2

A₃₁ = 6, A₃₂ = 2, A₃₃ = 5

∴ adj B = `[(3,2,6),(1,1,2),(2,2,5)]`

Now, B⁻¹ = `1/|B|` adj B

∴ B⁻¹ = `[(3,2,6),(1,1,2),(2,2,5)]`

∴ (AB)⁻¹ = B⁻¹A⁻¹

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

= `[(9-30+30,-3+12-12,3-10+12),(3-15+10,-1+6-4,1-5+4),(6-30+25,-2+12-10,2-10+10)]`

= `[(9,-3,5),(-2,1,0),(1,0,2)]`