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If A is a 2xx2 matrix such that A^(2)-4A...

If A is a `2xx2` matrix such that `A^(2)-4A+3I=O`, then prove that `(A+3I)^(-1)=7/24 I-1/24 A`.

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We have, `A^(2)-4A+3I=O`
`implies A(A+3I)-7A+3I=O`
`implies A(A+3I)-7(A+3I)=-24 I`
`implies (A+3I) (A-7I)=-24 I`
`implies (A+3I) (7/24 I-A/24)=I`
`implies (A+3 I)^(-1) =7/24 I- A/24`
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CENGAGE-MATRICES-Exercise 13.5
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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. If A=[{:(0,1,1),(1,0,1),(1,1,0):}] show that A^(-1)=(1)/(2)(A^(2)=3I)

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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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