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Question
Using matrix method, solve the system of equations
3x + 2y – 2z = 3, x + 2y + 3z = 6, 2x – y + z = 2.
Solution
Given system of equation is:
3x + 2y – 2z = 3
x + 2y + 3z = 6
And 2x – y + z = 2
or AX = B
i.e. `[(3, 2, -2),(1, 2, 3),(2, -1, 1)] [(x),(y),(z)] = [(3),(6),(2)]`
∴ X = `A^-1"B"`
For A–1
Co-factors are
A11 = 5
A12 =5
A13 = –5
A21 = 0
A22 = 7
A23 = 7
A31 = 10
A32 = –11
A33 = 4
∴ adj A = `[(5, 5, -5),(0, 7, 7),(10, -11, 4)]^"T"`
= `[(5, 0, 10),(5, 7, -11),(-5, 7, 4)]`
|A| = 3(5) + 2(5) + (–2)(–5) = 35
∴ A–1 = `("adj A")/|"A"|`
= `1/35 [(5, 0, 10),(5, 7, -11),(-5, 7, 4)]`
Now X = A–1B
⇒ `[(x),(y),(z)] = 1/35 [(5, 0, 10),(5, 7, -11),(-5, 7, 4)] [(3),(6),(2)]`
= `1/35 [(15 + 20),(15 + 42 - 22),(-5 + 42+ 8)]`
= `1/35 [(35),(35),(35)]`
= `[(1),(1),(1)]`
∴ x = 1, y = 1 and z = 1
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