Question

Find the inverse of the matrices (if it exists).

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

Answer 1

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

|A| = `|(1,0,0),(3,3,0),(5,2,-1)|`

= −1[−3 − 0] 

= 1 × (−3)

= −3

A₁₁ = `(-1)^(1 + 1) |(3,0),(2,-1)|`

= (−1)² [−3 − 0]

= 1 × (−3)

= −3

A₁₂ = `(-1)^(1 + 2) |(3,0),(5,-1)|`

= (−1)³ [−3 − 0]

= −1 × (−3)

= 3

A₁₃ = `(-1)^(1 + 3) |(3,3),(5,2)|`

= (−1)⁴ [6 − 15]

= 1 × (−9)

= −9

A₂₁ = `(-1)^(2 + 1) |(0,0),(2,-1)|`

= (−1)³ [0 − 0]

= 0

A₂₂ = `(-1)^(2 + 2) |(1,0),(5,-1)|`

= (−1)⁴ [−1 − 0]

= 1 × (−1)

= −1

A₂₃ = `(-1)^(2 + 3) |(1,0),(5,2)|`

= (−1)⁵ [2 − 0]

= −1 × 2

= −2

A₃₁ = `(-1)^(3 + 1) |(0,0),(3,0)|`

= (−1)⁴ [0 − 0]

= 0

A₃₂ = `(-1)^(3 + 2) |(1,0),(3,3)|`

= (−1)⁵ [0 − 0]

= 0

A₃₃ = `(-1)^(3 + 3) |(1,0),(3,3)|`

= (−1)⁶ [3 − 0]

= 1 × 3

= 3

∴ adj A = `[(-3,3,-9),(0,-1,-2),(0,0,3)] = [(-3,0,0),(3,-1,0),(-9,-2,3)]`

A⁻¹ = `1/|A|` adj A

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