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If A=[(1, 0),(-1 ,7)] , find k such th...

If `A=[(1, 0),(-1 ,7)]` , find `k` such that `A^2-8A+k I=O` .

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Given that,
`A=[(1, 0),(-1, 7)]`
we have to find `k` such that,
`A^2-8A+kI=0`
so,`A^2=[(1, 0),(-1, 7)]xx[(1, 0),(-1, 7)]`
`=[(1,0),(-8,49)]`
`A^2-8A+kI=0`
`implies[(1,0),(-8,49)]-8[(1, 0),(-1, 7)]+[(k,0),(0,k)]=[(0,0),(0,0)]`
...
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RD SHARMA-ALGEBRA OF MATRICES-Solved Examples And Exercises
  1. If A=[(3, 1),(-1, 2)] , show that A^2-5A+7I=O . Use this to find A^4

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  2. If A=[(3,-2) ,(4,-2)] , find k such that A^2-k A-2I2=0 .

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  3. If A=[(1, 0),(-1 ,7)] , find k such that A^2-8A+k I=O .

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  4. If A=[(1 ,2 ),(2, 1)] and f(x)=x^2-2x-3 , show that f(A)=O .

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  5. If A=[(2, 3),( 1, 2)] and I=[(1, 0),( 0, 1)] , then find lambda,mu ...

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  6. Find the value of x for which the matrix product [(2, 0, 7),( 0, ...

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  7. Solve the matrix equations: [(0, 2, 1)][(1, 2, 0),( 2 ,0 ,1 ),(1 ,0...

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  8. Solve the matrix equations: [(x,-5,-1)][(1, 0, 2) ,( 0,2 ,1),( 2, 0,...

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  9. If A=[(1, 2, 0 ),(3,-4, 5),( 0,-1, 3)] , compute A^2-4A+3I3 .

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  10. If f(x)=x^2-2x , find f(A), where A=[(0, 1, 2),( 4 ,5, 0 ),(0, 2 ,3)...

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  11. If f(x)=x^3+4x^2-x , find f(A) , where A=[(0 ,1, 2 ),(2,-3 ,0),( 1,-...

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  12. If A=[(1 ,0, 2), (0, 2, 1), (2, 0, 3)] , then show that A is a root of...

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  13. If A=[(1, 2 ,2 ),(2 ,1 ,2),( 2 ,2, 1)] , then prove that A^2-4A-5I=O ...

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  14. If A=[(3, 2, 0),( 1, 4 ,0 ),(0 ,0 ,5)] , show that A^2-7A+10 I3=O .

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  15. Without using the concept of inverse of a matrix, find the matrix [...

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  16. Find the matrix A such that : [(1 ,0 ),(0 ,1)]A=[(3 ,3 ,5 ),(1, 0 ,1)]

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  17. Find the matrix A such that : [2, 1, 3][(-1,0,-1),(-1,1 ,0 ),(0 ,1, 1...

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  18. Find a 2xx2 matrix A such that A[(1,-2),( 1, 4)]=6I2 .

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  19. If A=[(0, 0), (4, 0)] , find A^(16) .

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  20. If A=[(0,-x),(x,0)] , B=[(0 ,1 ),(1 ,0)] and x^2=-1 , then show tha...

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