Question

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

`[(1,2,3),(0,2,4),(0,0,5)]`

Answer 1

Let A = `[(1,2,3),(0,2,4),(0,0,5)]`

|A| = `|(1,2,3),(0,2,4),(0,0,5)|`

= 1(10 − 0) − 2(0 − 0) + 3(0 − 0)

= 10 ≠ 0

Now for cofactor C_ij = (−1)^i+j M_ij

C₁₁ = `(-1)^(1+1) |(2,4), (0,5)|` = 10

C₁₂ = `(-1)^(1+2) |(0,4), (0,5)|` = 0

C₁₃ = `(-1)^(1+3)|(0,2),(0,0)|` = 0

C₂₁ = `(-1)^(2+1) |(2,3), (0,5)|` = −10

C₂₂ = `(-1)^(2+2) |(1,3), (0,5)|` = 5

C₂₃ = `(-1)^(2+3) |(1,2), (0,0)|` = 0

C₃₁ = `(-1)^(3+1) |(2,3), (2,4)|` = 2

C₃₂ = `(-1)^(3+2) |(1,3), (0,4)|` = −4

C₃₃ = `(-1)^(3+3)|(1,2), (0,2)|` = 2

∴ Adj A = `[(10,0,0),(-10,5,0),(2,-4,2)]^' = [(10,-10,2),(0,5,-4),(0,0,2)]`

Hence, A⁻¹ = `1/|A|` Adj A

= `1/10 [(10,-10,2),(0,5,-4),(0,0,2)]`