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Question
if A = `((2,3,10),(4,-6,5),(6,9,-20))`, Find `A^(-1)`. Using `A^(-1)` Solve the system of equation `2/x + 3/y +10/z = 2`; `4/x - 6/y + 5/z = 5`; `6/x + 9/y - 20/z = -4`
Solution
Let 1/x be a, 1/y be b and 1/x be c
Here
A = `[(2,3,10),(4,-6,5),(6,9,-20)]` and B = `[(2),(5),(-4)]`
|A| = `[(2,3,10),(4,-6,5),(6,9,-20)]`
= 2(120 - 45) -3 (-80-30) +10 (36 + 36)
= 150 + 330 + 720
= 1200
Let `C_(ij)` be the cofactors of the elements `a_(ij)` in `A[a_(ij)]` Then
`X = A^(-1)B`
`=>[(a),(b),(c)] = 1/1200 [(75,150,75),(110,-100,30),(72,0,-24)][(2),(5),(-4)]`
`=> [(a),(b),(c)] = 1/1200 [(150+750-300),(220-500-120),(144+0+96)]`
`=> [(a),(b),(c)] = 1/1200 [(600),(-400),(240)]`
`=> [(a),(b),(c)] = [(1/2),(1/(-3)), (1/5)]`
`=> x = 1/a = 2, y = 1/b = -3, " and " z = 1/c = 5`
∴ x = 2, y = 3 and z = 5
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