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Prove by Mathematical Induction that `(A)^(n)=(A^(n))` where `n inN` for any square matrix A.

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Prove by mathematical induction that sum_(r=0)^(n)r^(n)C_(r)=n.2^(n-1), forall n in N .

Show using mathematical induciton that n!lt ((n+1)/(2))^n . Where n in N and n gt 1 .

Using mathematical induction prove that (d)/(dx) (x^(n))= n x^(n-1) for all positive integers n.

prove using mathematical induction, n(n+1)(n+5) is divisible by 6 for all natural numbers

Use the principle of mathematical induction to show that (a^(n) - b^n) is divisble by a-b for all natural numbers n.

Use the principle of mathematical induction to show that 5^(2n+1)+3^(n+2).2^(n-1) divisible by 19 for all natural numbers n.

If A and B are square matrices of the same order such that AB=Ba , then prove by inducation that AB^(n)=B^(n)A . Further , prove that (AB)^(n)=A^(n)B^(n) for all n in N .

If A=[{:(1,alpha),(0,1):}] then by the principle of mathematical induction . Prove that A^(n)=[{:(1,nalpha),(0,1):}],AAn inN .

Prove each of the statements by the principle of mathematical induction : n(n^2 + 5) is divisible by 6, for each natural number n.

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KUMAR PRAKASHAN-MATRICES -Solutions of NCERT Exemplar Problems (Short Answer Type Questions)
  1. Show that A'A and A A ' are both symmetric matrices for any matrix A .

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  2. Let A and B be square matrices of the order 3xx3 . Is (AB)^(2)=A^(2)B^...

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  3. Show that if A and B are square matrics such that AB=BA , then (A+B)^(...

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  4. Let A=[{:(1,2),(-1,3):}],B=[{:(4,0),(1,5):}],C=[{:(2,0),(1,-2):}]and a...

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  5. If A=[{:(cosq,sinq),(-sinq,cosq):}] , then show that A^(2)=[{:(cos(2q...

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  6. If A=[{:(0,-x),(x,0):}],B=[{:(0,1),(1,0):}]andx^(2)=-1 then show that ...

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  7. Verify that A^(2)=I when A=[{:(0,1,-1),(4,-3,4),(3,-3,4):}].

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  8. Prove by Mathematical Induction that (A)^(n)=(A^(n)) where n inN for a...

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  9. Find inverse , by elementary row operations (if possible), of the foll...

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  10. If [{:(xy,4),(z+6,x+y):}]=[{:(8,w),(0,6):}] , then find values of x,y,...

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  11. If A=[{:(1,5),(7,12):}],B=[{:(9,1),(7,8):}] find a matrix C such that ...

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  12. If A=[{:(3,-5),(-4,2):}], then find A^(2)-5A-14I. Hence , obtain A^(3)...

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  13. Find the values of a,b ,c and d, if 3[{:(a,b),(c,d):}]=[{:(a,6),(-1,...

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  14. Find the matrix X such that , [{:(2,-1),(1,0),(-3,4):}]X=[{:(-1,-8...

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  15. If A=[{:(1,2),(4,1):}], find A^(2)+2A+7I.

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  16. If A=[{:(cosalpha,sinalpha),(-sinalpha,cosalpha):}],andA^(-1)=A', find...

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  17. If the maxtrix [{:(0,a,3),(2,b,-1),(c,1,0):}] is a skew symmetric mat...

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  18. If p(x)=[{:(cosx,sinx),(-sinx,cosx):}] then show that p(x)*p(y)=p(x+y)...

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  19. If A is square matrix such that A^(2)=A, show that (I+A)^(3)=7A+I.

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  20. If A,B are square matrices of same order and B is a skew -symmetric ma...

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