Question

Find the inverses of the following matrices by the adjoint method:

`[(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) - 0 + 0

= 1(10) - 0 + 0

= 10 ≠ 0

∴ A^-1 exists.

First we have to find the co-factor matrix

`= ["A"_"ij"]_(3xx3)`, where `"A"_"ij" = (-1)^("i"+"j") "M"_"ij"`

Now, A₁₁ = `(- 1)^(1 + 1) "M"_11 = |(2,4),(0,5)| = 10 - 0 = 10`

A₁₂ = `(- 1)^(1 + 2) "M"_12 = - |(0,4),(0,5)| = - 0 - 0 = 0`

A₁₃ = `(- 1)^(1 + 3) "M"_13 = |(0,2),(0,0)| = 0 - 0 = 0`

A₂₁ = `(- 1)^(2 + 1) "M"_21 = - |(2,3),(0,5)| = - 10 - 0 = - 10`

A₂₂ = `(- 1)^(2 + 2) "M"_22 = |(1,3),(0,5)| = 5 - 0 = 5`

A₂₃ = `(- 1)^(2 + 3) "M"_23 = - |(1,2),(0,0)| = - 0 - 0 = 0`

A₃₁ = `(- 1)^(3 + 1) "M"_31 = |(2,3),(2,4)| = 8 - 6 = 2`

A₃₂ = `(- 1)^(3 + 2) "M"_32 = - |(1,3),(0,4)| = - 4 - 0 = - 4`

A₃₃ = `(- 1)^(3 + 3) "M"_33 =  |(1,2),(0,2)| = 2 - 0 = 2`

∴ the co-factor matrix

`= [("A"_11,"A"_12,"A"_13),("A"_21,"A"_22,"A"_23),("A"_31,"A"_32,"A"_33)] = [(10,0,0),(-10,5,0),(2,-4,2)]`

∴ adj A = `[(10,-10,2),(0,5,-4),(0,0,2)]`

∴ A^-1 = `1/|"A"|`(adj A)

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

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