Suppose `A` and `B` are two non singular matrices such that `B != I, A^6 = I` and `AB^2 = BA`. Find the least value of `k` for `B^k = 1`

Explanation

Given, `AB = BA^(2) rArr B= A^(-1) BA^(2) rArr B^(3) = I`
`rArr (A^(-1) BA A) ( A^(-1) BA A ) (A^(-1) BA A) = I`
`rArr (A^(-1) BA A) (BA ) ( BA A) = I [because A^(-1) A=I]`
`rArr A^(-1) B (BA^(2)) (BA^(2)) A A=I`
`rArr A^(-1) B(BA^(2)) (BA^(2)) A A = I [ because AB =BA^(2) ] `
`rArr A^(-1) BBA (AB) A^(4) = I`
`rArr A^(-1) BBA (BA^(2) )A^(4) = I [because AB = BA^(2)]`
`rArr A^(-1) BB (AB) A^(6) = I`
`rArr A^(-1) BB(BA^(2)) A^(6) = I [because AB = BA^(2)]`
`rArr A^(-1) B^(3) A^(8) = I`
` rArr (A^(-1)I) A^(8) = I [ B^(3) = I]`
`rArr A^(-1) A^(8) = I`
`rArr A^(7) = I = A^(k) [ because A^(k) = I]`
`rArr A^(k) = A^(7)`
`therefore` Least value of `k` is 7.