Question

If A = `[(2,-3,5),(3,2,-4),(1,1,-2)]` find A⁻¹. Using A⁻¹ solve the system of equations:

2x – 3y + 5z = 11

3x + 2y – 4z = –5

x + y – 2z = –3

Answer 1

|A| = `|(2,-3,5),(3,2,-4),(1,1,-2)|`

= 2[2 × (−2) − 1 × (−4)] − (−3)[3 × (−2) − (1) × (−4)] + 5[3 × 1 − 1 × 2]

= 2[−4 + 4] + 3[−6 + 4] + 5[3 − 2]

= 0 + 3 × (−2) + 5 × 1

= −6 + 5

= −1 ≠ 0

∴ A⁻¹ can be known,

Cofactors of the elements of |A|:

A₁₁ = `|(2,-4),(1,-2)|`

= −4 + 4

= 0

A₁₂ = `-|(3,-4),(1,-2)|`

= −(−6 + 4)

= 2

A₁₃ = `|(3,2),(1,1)|`

= 3 − 2

= 1

A₂₁ = `- |(-3,5),(1,-2)|`

= −(6 − 5)

= −1

A₂₂ = `|(2,5),(1,-2)|`

= −4 − 5

= −9

A₂₃ = `- |(2,-3),(1,1)|`

= −(2 + 3)

= −5

A₃₁ = `|(-3,5),(2,-4)|`

= 12 − 10

= 2

A₃₂ = `-|(2,5),(3,-4)|`

= −(−8 − 15)

= 23

A₃₃ = `|(2,-3),(3,2)|`

= 4 + 9

= 13

The cofactor matrix of the elements of |A| = `[(0,2,1),(-1,-9,-5),(2,23,13)]`

∴ adj A = `[(0,2,1),(-1,-9,-5),(2,23,13)] = [(0,-1,2),(2,-9,23),(1,-5,13)]`

∴ A⁻¹ = `1/|A|` adj A

= `1/(-1)[(0,-1,2),(2,-9,23),(1,-5,13)]`

= `[(0,1,-2),(-2,9,-23),(-1,5,-13)]`

Writing the given equation in the form AX = B,

Or A = `[(2,-3,5),(3,2,-4),(1,1,-2)]`, X = `[(x),(y),(z)]`, B = `[(11),(-5),(-3)]`

∴ X = A⁻¹B

`[(x),(y),(z)] = [(0,1,-2),(-2,9,-23),(-1,5,-13)] [(11),(-5),(-3)]`

= `[(0 - 5 + 6),(-22 - 45 + 69),(-11 - 25 + 39)]`

= `[(1),(2),(3)]`

⇒ x = 1, y = 2, z = 3