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If B ,C are square matrices of order na ...

If `B ,C` are square matrices of order `na n difA=B+C ,B C=C B ,C^2=O ,` then without using mathematical induction, show that for any positive integer `p ,A^(p-1)=B^p[B+(p+1)C]` .

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`because A= B+CrArr A^(p+1) = (B+C)^(p+1) `
`= ""^(p+1) C_(0) B^(p+1) + ""^(p=1) C_(1) B^(p) C + ""^(P+1) C_(2) B^(p-1) C^(2) +... + ""^(p+1)C_(p+1) C^(p+1)`
`= B^(p+1)+ ""^(p+1) C_(1) B^(p) C + 0 + 0 +... [because C^(2) = 0 rArr C^(2) = C^(3) =... = 0]`
`= B^(P) [B+(p+1) C]`
Hence, `A^(p+1) = B^(p)[B+(p+1)C]`
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ARIHANT MATHS-MATRICES -Exercise (Questions Asked In Previous 13 Years Exam)
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  2. Let A=[(1,0,0),(0,1,1),(0,-2,4)],I=[(1,0,0),(0,1,0),(0,0,1)] and A^-1=...

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  3. Evluate int 3x^2 dx

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  4. If A=[(1,0),(1,1)] and I=[(1,0),(0,1)] then which one of the following...

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  5. If A^(2)-A+I=O, then A^(-1) is equal to

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  6. Let {:A=[(1,0,0),(2,1,0),(3,2,1)]:}and U1,U2,U3 be column matrices sat...

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  7. Let A = [(1,0,0), (2,1,0), (3,2,1)], and U1, U2 and U3 are columns of ...

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  8. If A= ((1,0,0),(2,1,0),(3,2,1)), U(1), U(2), and U(3) are column matri...

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  9. Let A=[{:(1,2),(3,4):}]and B = [{:(a,0),(0,b):}] where a, b are natura...

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  10. If A and B are square matrices of size nxxn such that A^2-B^2 = (A-B)(...

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  11. Let A= [[5,5alpha,alpha],[0,alpha,5alpha],[0,0,5]] . If |A^2|...

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  12. Let A and B be 3xx3 matrtices of real numbers, where A is symmetric, "...

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  13. Let A be a square matrix all of whose entries are integers. Then wh...

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  14. Let A be a 2xx2 matrix with real entries. Let I be the 2xx2 identi...

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  15. Let A be the set of all 3xx3 symmetric matrices all of whose either 0 ...

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  16. Let A be the set of all 3xx3 symmetric matrices all of whose either 0 ...

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  17. The number of 3xx3 matrices A whose are ether 0 or 1 and for which t...

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  18. Let A be a 2xx2 matrix Statement -1 adj (adjA)=A Statement-2 abs(a...

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  19. The number of 3xx3 matrices a whose entries are either 0 or 1 and for ...

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  20. Let P be an odd prime number and T(p) be the following set of 2xx2 mat...

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  21. Let p be an odd prime number and T(P) be the following set of 2xx2 m...

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