Question

Solve by matrix inversion method:

3x – y + 2z = 13; 2x + y – z = 3; x + 3y – 5z = - 8

Answer 1

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

[A_ij] = `[(-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.