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If A=[(2, 3),(-1 ,0)] , show that A^2-2...

If `A=[(2, 3),(-1 ,0)]` , show that `A^2-2A+3I_2=O` .

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Given,
`A=[(2, 3),(-1 ,0)]`
`A^2=A*A`
`=[(2, 3),(-1 ,0)][(2, 3),(-1 ,0)]`
`=[(4-3, 6+0),(-2+0 ,-3+0)]`
`=[(1, 6),(-2 ,-3)]`
we know that `I=[(1,0),(0,1)]`
now,
`A^2-2A+3I_2`
`=[(1, 6),(-2 ,-3)]-2[(2, 3),(-1 ,0)]+3[(1,0),(0,1)]`
`=[(1, 6),(-2 ,-3)]-[(4, 6),(-2 ,0)]+[(3,0),(0,3)]`
`=[(0,0),(0,0)]`
Hence
`A^2-2A+3I_2=0`
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RD SHARMA-ALGEBRA OF MATRICES-Solved Examples And Exercises
  1. If A=[(3 ,1),(-1, 2)] and I=[(1, 0), (0 ,1)] , then find lambda so tha...

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  2. If A=[[3 ,1],[-1, 2]] , show that A^2-5A+7I2=O .

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  3. If A=[(2, 3),(-1 ,0)] , show that A^2-2A+3I2=O .

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  4. Show that the matrix A=[(2, 3), (1, 2)] satisfies the equation A^3-4A...

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  5. Show that the matrix A=[(5, 3), (12 ,7)] is a root of the equation A...

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  6. If A=[(3,-5),(-4, 2)] , find A^2-5A-14 Idot

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  7. If A=[(3, 1),(-1, 2)] , show that A^2-5A+7I=O . Use this to find A^4

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

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

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

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

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

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

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

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

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

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

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

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

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

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