InterviewSolution
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InterviewSolution
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Suppose A and B be two ono-singular matrices such that `AB= BA^(m), B^(n) = I and A^(p) = I `, where `I` is an identity matrix. If `m = 2 and n = 5 ` then p equals to |
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Answer» `AB=BA^2` `B^-1(AB)= B^-1(BA^2)` `= (B^-1B)A^2` `= A^2` `A^2 = B^-1AB` `A^4= A^2A^2= (B^-1AB)(B^-1AB) = B^-1ABB^-1AB` `= B^-1A(BB^-1)AB` `= B^-1A(I)AB` `=B^-1A^2B` `=B^-1(B^-1AB)B` `=B^-2AB^2` `A^8=A^4A^4= (B^-2AB^2)(B^-2AB^2)` `= B^-2A^2B^2` `=B^-2(B^-1AB)B^2` `= B^-3AB^3` `A^16= A^8A^8 = B^-3B^3B^-3AB^3` `= B^-3A^2B^3` `=B^-3(B^-1AB)B^3` `= B^-4AB^4` `A^32= A^16A^16 = B^-4AB^4B^-4AB^4` `= B^-4A^2B^4` `= B^-4(B^-1AB)B^4` `=B^-5AB^5` as, `B^5 = I ` so, `B^-5 = I` putting in eqn `=IAI` `A^32=A` multilpying by `A^-1` to both sides `A^32 = A` `A^32A^-1 = A*A^-1` `A^31 = I` |
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