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Question
x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1
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Solution
x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1
This system can be written as AX = B,
`A = [(1,1,1),(2,-1,1),(1,-2,3)]`
`x = [(x),(y),(z)]`
`B = [(3),(2),(2)]`
|A| = 1(-3+2) –1 (6 - 1) + 1(-4 + 1)
= -1 - 5 - 3
= -9
`A_11 = 1 |(-1,1),(-2,3)| = -1`
`A_12 = -1|(2,1),(1,3)| = -5`
`A_13 = 1|(2,-1),(1,-2)| = -3`
`A_21 = -1 |(1,1),(-2,3)| = -5`
`A_22 = 1|(1,1),(1,3)| = 2`
`A_23 = -1 |(1,1),(1,-2)| = 3`
Hence adj A = `[(A_11,A_21,A_31),(A_12,A_22,A_32),(A_13,A_23,A_33)]`
`=[(-1,-5,2),(-5,2,1),(-3,3,-3)]`
`A^-1 = 1/|A| adjA`
`=1/-9[(-1,-5,2),(-5,2,1),(-3,3,-3)]`
`X = A^-1B`
`X = 1/-9 [(-1,-5,2),(-5,2,1),(-3,3,-3)][(3),(2),(2)]`
`[(x),(y),(z)] = - (-1)/9 [(-3-10+4),(-15+4+2),(-9+6-6)]`
`[(x),(y),(z)] = [((-9)/-9),((-9)/-9),((-9)/-9)] = [(1),(1),(1)]`
x = 1, y = 1, z = 1
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