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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
Solution
`[(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
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