Question

If A = `[(1, 1, 1),(0, 1, 3),(1, -2, 1)]`,find A^-1

hence, solve the following system of equations

x + y + z = 6
y + 3z =11
x- 2y + z = 0

Answer 1

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

Cofactors

A₁₁ = 7, A₁₂ = 3, A₁₃ = -1
A₂₁ = -3, A₂₂ = 0, A₂₃ = 3
A₃₁ = 2, A₃₂ = -3, A₃₃ = 1

A^-1 = `(Adj("A")) /|"A"|`

Adj (A) = `[(7, 3, -1),(-3, 0, 3),(2, -3, 1)]^"T" = [(7, -3, 2),(3, 0, -3),(-1, 3, 1)] `

|A| = 9

`"A"^-1 = 1/9 [(7, -3, 2),(3, 0, -3),(-1, 3, 1)] `

For system of equations 

AX = B

X = `"A"^-1 "B"`

`[("x"),("y"),("z")] = 1/9 [(7, -3, 2),(3, 0, -3),(-1, 3, 1)] [(6),(11),(0)]`

`[("x"),("y"),("z")] = 1/9 [(9),(18),(27)]`

x =1, y = 2, z = 3