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Suppose A and B be two ono-singular matr...

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

Text Solution

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`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`
...
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