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By the method of matrix inversion, solve...

By the method of matrix inversion, solve the system.
`[(1,1,1),(2,5,7),(2,1,-1)][(x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3))]=[(9,2),(52,15),(0,-1)]`

Text Solution

Verified by Experts

The correct Answer is:
`x_(1)=1, x_(2)=3, x_(3)=5` or `y_(1)=-1, y_(2)=2, y_(3)=1`
We have
`[(1,1,1),(2,5,7),(2,1,-1)] [(x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3))]=[(9,2),(52,15),(0,-1)]`
`implies AX=B` (1)
Clearly `|A|=-4 ne 0`. Therefore,
`:." adj A"=[(-12,16,-8),(2,-3,1),(2,1,3)]^(T)=[(-12,2,2),(16,-3,-5),(-8,1,3)]`
`:. A^(-1)=("adj. A").(|A|)=(-1)/4 [(-12,2,2),(16,-3,-5),(-8,1,3)]`
Now, `A^(-1)B=(-1)/4 [(-12,2,2),(16,-3,-5),(-8,1,3)][(9,2),(52,15),(0,-1)]`
`=(-1)/4 [(-4,4),(-12,-8),(-20,-4)]=[(1,-1),(3,2),(5,1)]`
From Eq. (1), we get
`X=A^(-1) B`
`implies [(x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3))]=[(1,-1),(3,2),(5,1)]`
`implies x_(1)=1, x_(2)=3, x_(3)=5`
or `y_(1)=-1, y_(2)=2, y_(3)=1`
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