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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)]`

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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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CENGAGE-MATRICES-Exercise 13.5
  1. By the method of matrix inversion, solve the system. [(1,1,1),(2,5,7...

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  2. Let A=[[2,0,7] , [0,1,0], [1,-2,1]] and B=[[-x,14x,7x] , [0,1,0] , [x,...

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  3. Find A^(-1) if A=|(0,1,1),(1,0,1),(1,1,0)| and show that A^(-1)=(A^(2)...

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  4. For the matrix A=[3 1 7 5] , find x and y so that A^2+x I=y Adot

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  5. If A^(3)=O, then prove that (I-A)^(-1) =I+A+A^(2).

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  6. If A=[[cos alpha, -sin alpha] , [sin alpha, cos alpha]], B=[[cos2beta,...

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  7. If A=[(1,2,2),(2,2,3),(1,-1,3)], C=[(2,1,1),(2,2,1),(1,1,1)], D=[(10),...

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  8. If A is a 2xx2 matrix such that A^(2)-4A+3I=O, then prove that (A+3I)^...

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  9. For two unimobular complex numbers z(1) and z(2), find [(bar(z)(1),-z(...

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  10. Prove that inverse of a skew-symmetric matrix (if it exists) is skew-s...

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  11. If square matrix a is orthogonal, then prove that its inverse is also ...

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  12. If A is a skew symmetric matrix, then B=(I-A)(I+A)^(-1) is (where I is...

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  13. Prove that ("adj. "A)^(-1)=("adj. "A^(-1)).

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  14. Using elementary transformation, find the inverse of the matrix A=[(a,...

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  15. Show that the two matrices A, P^(-1) AP have the same characteristic r...

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  16. Show that the characteristics roots of an idempotent matris are either...

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  17. If alpha is a characteristic root of a nonsin-gular matrix, then prove...

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