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Question
The sum of three numbers is 20. If we multiply the first by 2 and add the second number and subtract the third we get 23. If we multiply the first by 3 and add second and third to it, we get 46. By using the matrix inversion method find the numbers.
Solution
Let the three numbers be x, y, and z.
x + y + z = 20
2x + y – z = 23
3x + y + z = 46
The given system can be written as
`[(1,1,1),(2,1,-1),(3,1,1)][(x),(y),(z)] = [(20),(23),(46)]`
AX = B
where A = `[(1,1,1),(2,1,-1),(3,-1,1)]`. X = `[(x),(y),(z)]` and B = `[(20),(23),(46)]`
A = `[(1,1,1),(2,1,-1),(3,1,1)]`
|A| = `|(1,1,1),(2,1,-1),(3,1,1)|`
`= 1|(1,-1),(1,1)| -1|(2,-1),(3,1)| + 1|(2,1),(3,1)|`
= 1[1 + 1] - 1(2 + 3) + 1(2 - 3)
= 1(2) - 1(5) + 1(-1)
= 2 - 5 - 1 = - 4
[Aij] = `[(2,-5,-1),(-|(1,1),(1,1)|,|(1,1),(3,1)|,-|(1,1),(3,1)|),(|(1,1),(1,-1)|,-|(1,1),(2,-1)|,|(1,1),(2,1)|)]`
`= [(2,-5,-1),(-(0),(1-3),-(1-3)),((-1-1),-(-1-2),(1-2))] = [(2,-5,-1),(0,-2,2),(-2,3,-1)]`
adj A = [Aij]T = `[(2,0,-2),(-5,-2,3),(-1,2,-1)]`
`"A"^-1 = 1/|"A"|`(adj A)
`= 1/(-4)[(2,0,-2),(-5,-2,3),(-1,2,-1)]`
X = A-1B
`[(x),(y),(z)] = 1/(-4)[(2,0,-2),(-5,-2,3),(-1,2,-1)] [(20),(23),(46)]`
`= (-1)/(4) [(40+0-92),(-100-46+138),(-20+46-46)]`
`= (-1)/4[(-52),(-8),(-20)]`
`[(x),(y),(z)] = [(13),(2),(5)]`
The numbers are 13, 2, and 5.
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