Let A and B be two non-singular matrices such
that `A ne I, B^(2) = I and AB = BA^(2)` , where I is the identity
matrix, the least value of k such that ` A^(k) = I 1 is

Options

A. `p = mn^(2)`
B. `p=m^(n)-1`
C. `p=n^(m) - 1`
D. `p = m^(n-1)`

Answer

B

Explanation

`because AB = BA^(m) `
` rArr B = A^(-1) BA^(m)`
`therefore B^(n) = underset("n times")(underbrace((A^(-1) BA^(m))(A^(-1)BA^(m))... (A^(-1) BA^(m))))`
`=A^(-1) underset("n times")(underbrace(BA^(m-1)BA^(m-1)... BA^(m-1)BA^(m-1)))A` ...(i)
Given, ` AB = BA^(m)`
` rArr A AB = ABA^(m) = BA^(2m) rArr A A AB = BA^(3m)`
Similarly, `A^(x) B = BA^(mx) AA m in N`
From Eq. (i) we get
`B^(n)=A^(-1) BA^(m-1) underset("(n-1) times")(underbrace(BA^(m-1)BA^(m-1)... BA^(m-1)BA^(m-1)))A`
`=A^(-1) B(A^(m-1) B)A^(m-1) underset("(n-2) times")(underbrace(BA^(m-1)... BA^(m-1)BA^(m-1)))A`
`=A^(-1) BBA^((m-1)m) A^(m-1) underset("(n-2) times")(underbrace(BA^(m-1)... BA^(m-1)BA^(m-1)))A`
`=A^(-1) B^(2)A^((m^(2)-1)) underset("(n-2) times")(underbrace(BA^(m-1)... BA^(m-1)BA^(m-1)))A`
`..." " ..." "... " "... `
`=A^(-1) B^(m) (A) ^(m^(n)-1)A`
`I = A^(-1) I A^(m^(n)-1) A [because B^(n) = I]`
`I = A^(-1) A^(m^(n)-1) A= A^(-1) A^(m^(n))`
`rArr I = A^(m^(n)-1)`
`therefore p= m^(n)-1 " "...(ii) [because A^(p) = I]`
From Eq. (ii), we get
`p = m^(n) - 1`