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Question
Solve by matrix inversion method:
3x – y + 2z = 13; 2x + y – z = 3; x + 3y – 5z = - 8
Solution
The given system can be written as
`[(3,-1,2),(2,1,-1),(1,3,-5)][(x),(y),(z)] = [(13),(3),(-8)]`
AX = B
Where A = `[(3,-1,2),(2,1,-1),(1,3,-5)]`, X = `[(x),(y),(z)]` and B = `[(13),(3),(-8)]`
|A| = `|(3,-1,2),(2,1,-1),(1,3,-5)|`
= 3(-5 + 3) – (-1) (-10 + 1) + 2 (6 – 1)
= 3(-2) + 1(-9) + 2(5)
= - 6 – 9 + 10
= - 5
[Aij] = `[(-2,-(-9),5),(-|(-1,2),(3,-5)|,|(3,2),(1,-5)|,-|(3,-1),(1,3)|),(|(-1,2),(1,-1)|,-|(3,2),(2,-1)|,|(3,-1),(2,1)|)]`
`= [(-2,9,5),(-(5-6),(-15-2),-(9+1)),((1-2),-(-3-4),(3+2))] => [(-2,9,5),(1,-17,-10),(-1,7,5)]`
adj A = `["A"_"ij"]^"T" = [(-2,1,-1),(9,-17,7),(5,-10,5)]`
`"A"^-1 = 1/|"A"|`(adj A)
`= 1/(-5)[(-2,1,-1),(9,-17,7),(5,-10,5)]`
X = A-1B
`= 1/(-5)[(-2,1,-1),(9,-17,7),(5,-10,5)][(13),(3),(-8)]`
`=> 1/(-5)[(-26+3+8),(117-51-56),(65-30-40)]`
`=> 1/(-5)[(-15),(10),(-5)]`
`[(x),(y),(z)] = [(3),(-2),(1)]`
∴ x = 3, y = -2, z = 1.
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