Question

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

`[(1,-1,2),(0,2,-3),(3,-2,4)]`

Answer 1

Let A = `[(1,-1,2),(0,2,-3),(3,-2,4)]`

∴ |A| = `|(1,-1,2),(0,2,-3),(3,-2,4)|`

= 1(8 − 6) + 1(0 + 9) + 2(0 − 6)

= 2 + 9 − 12

= −1 ≠ 0

∴ A is invertible.

Let C_ij be the cofactor of a_ij in A; then the cofactors of elements of A are given by,

C₁₁ = `(-1)^(1+1) |(2,-3), (-2,4)|`

= 8 − 6

= 2

C₁₂ = `(-1)^(1+2) |(0,-3), (3,4)|`

= −(0 + 9)

= −9

C₁₃ = `(-1)^(1+3)|(0,2),(3,-2)|`

= 0 − 6

= −6

C₂₁ = `(-1)^(2+1) |(-1,2), (-2,4)|`

= −(−4 + 4)

= 0

C₂₂ = `(-1)^(2+2) |(1,2), (3,4)|`

= 4 − 6

= −2

C₂₃ = `(-1)^(2+3) |(1,-1), (3,-2)|`

= −(−2 + 3)

= −1

C₃₁ = `(-1)^(3+1) |(-1,2), (2,-3)|`

= 3 − 4

= −1

C₃₂ = `(-1)^(3+2) |(1,2), (0,-3)|`

= −(−3 − 0)

= 3

C₃₃ = `(-1)^(3+3)|(1,-1), (0,2)|`

= 2 + 0

= 2

∴ adj A = `[(2,-9,-6),(0,-2,-1),(-1,3,2)] = [(2,0,-1),(-9,-2,3),(-6,-1,2)]`

A⁻¹ = `1/|A|` adj A

= `1/-1[(2,0,-1),(-9,-2,3),(-6,-1,2)]`

= `[(-2,0,1),(9,2,-3),(6,1,-2)]`