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

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

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Given that,
`A= [(0 ,1, 2 ),(2,-3 ,0),( 1,-1 ,0)]`
so,`A^2= [(0 ,1, 2 ),(2,-3 ,0),( 1,-1 ,0)]xx [(0 ,1, 2 ),(2,-3 ,0),( 1,-1 ,0)]`
`=[(4,-5,0),(-6,11,4),(-2,4,2)]`
And `A^3=A^2xxA=[(4,-5,0),(-6,11,4),(-2,4,2)]xx [(0 ,1, 2 ),(2,-3 ,0),( 1,-1 ,0)]`
`implies A^3=[(-10,19,8),(26,-43,-12),(10,-16,-4)]`
`f(x)=x^3+4x^2-x`
`implies f(A)=A^3+4A^2-A`
...
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RD SHARMA-ALGEBRA OF MATRICES-Solved Examples And Exercises
  1. If A=[(1, 2, 0 ),(3,-4, 5),( 0,-1, 3)] , compute A^2-4A+3I3 .

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

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

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

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

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

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

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

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

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

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

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

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  13. If A=[(1, 0,-3 ),(2, 1 ,3 ),(0, 1 ,1)] , then verify that A^2+A=A(A+I...

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

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  15. If P(x)=[cosxsinx-sinxcosx] , then show that P(x)\ P(y)=P(x+y)=P(y)\ P...

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  16. If P=[(x,0, 0),( 0,y,0 ),(0, 0,z)] and Q=[(a,0 ,0 ),(0,b,0 ),(0, 0,c)...

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  17. If A=[(2, 0, 1 ),(2, 1, 3),( 1,-1, 0)] , find A^2-5A+4I and hence fi...

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  18. If A=[(1 ,1),( 0, 1)] , prove that A^n=[(1,n),(0, 1)] for all positive...

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  19. If A=[(a, b),(0, 1)] , prove that A^n=[(a^n ,b((a^n-1)/(a-1))),(0, 1)]...

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  20. If A=[(costheta,isintheta),(isintheta,costheta)] , then prove by pri...

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