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If A=[{:(costheta,sintheta),(-sintheta,...

If `A=[{:(costheta,sintheta),(-sintheta,costheta):}]`, then prove that `A^(n)=[{:(cosntheta,sinntheta),(-sinntheta,cosn theta):}], n inN`.

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The correct Answer is:
`A^(n)=[{:(cosn theta,sin n theta),(-sinn theta,cosn theta):}],AAn inN`.
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ARIHANT PUBLICATION-MATRICES -CHAPTER PRACTICE (Very Short Answer Type Questions)
  1. If A=[{:(costheta,sintheta),(-sintheta,costheta):}], then prove that ...

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  2. What possible orders of a matrix can have , if it has (i) 5 elemen...

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  3. Construct 2xx2 matrix , A=[a(ij)] whose elements are given by a(ij)=...

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  4. Find the number of all possible matrices of order 3xx3 with each of ...

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  5. If A = [ {:(cos alpha, - sin alpha),(sin alpha, cos alpha):}], then fo...

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  6. If [{:(x+3,z+4,2y-7),(-6,a-1,0),(b-3,-21,0):}]=[{:(0,6,3y-2),(-6,-3,2c...

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  7. If Possible , then find the sum of the matrices A and B , where A=[{:(...

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  8. Simplify[{:(cos^(2)x,sin^(2)x),(sin^(2)x,cos^(2)x):}]+[{:(sin^(2)x,cos...

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  9. Find the value of y and x from the following equation 2[{:(x,5),(7,y-...

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  10. Find non - zero values of x satisfying the matrix equation x[{:(2x,...

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  11. Solve for x and y , x [{:(2),(1):}]+y[{:(3),(5):}]=[{:(-8),(-11):}]

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  12. If A=[{:(2,4),(3,2):}]andB=[{:(1,3),(-2,5):}], then find A-B.

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  13. If [{:(9,-1,4),(-2,1,3),(1,2,-1):}]=A+[{:(1,2,-1),(0,4,9),(3,1,-2):}] ...

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  14. Assume Y, W and P are the matrices of orders 3xxk,nxx3andpxxk .Find t...

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  15. Let A=[a(ij)](nxxn) be a diagonal matrix whose diagonal elements are...

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  16. Suppose A=[{:(5,4),(2,3):}]andB=[{:(3,5,1),(6,8,4):}] , then find AB a...

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  17. Find x from the matrix equation [{:(1,3),(4,5):}][{:(x),(2):}]=[{:(5)...

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  18. If matrix A=[{:(1,-1),(-1,1):}]andA^(2)=kA , then write the value of k...

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  19. Show by an example that for AneO,BneOandAB=O.

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  20. If A=[{:(1,2),(3,4):}]then find A+A .

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  21. If A=[{:(0,-1,2),(4,3,-4):}]andB=[{:(4,0),(1,3),(2,6):}] then verify t...

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