Question

Suppose A and B be two non-singular matrices, such that $AB=B{A}^m$ and $B^n=I$ and $A^p=I$
Which of the following represents the correct relation between m, n and p?

• A. $m{n}^2=p$
• B. $p=m^n-1$
• C. $p=n^m-1$
• D. $p=m^{n-1}$

Answer 1

The correct option is B $p=m^n-1$

$B=A^{-1}B{A}^m{B}^n=\underset{n-times}{\underset{}{A^{-1}{A}^m{A}^{-1}B{A}^m\cdots A^{-1}B{A}^m}}{B}^n=A^{-1}\underset{n-times}{\underset{}{B{A}^{m-1}B{A}^{m-1}\cdots B{A}^{m-1}B{A}^{m-1}A}}$
We know $AB=B{A}^m$
$\Rightarrow AAB=AB{A}^m=B{A}^{2m}\Rightarrow AAAB=B{A}^{3m}$
Similarly $A\times B=B{A}^{mx}$
$\Rightarrow B^n=I=A^{-1}BB(A{)}^{(m-1)m}{A}^{m-1}\underset{(n-2)times}{\underset{}{B{A}^{m-1}B{A}^{m-1}\cdots B{A}^{m-1}A}}\Rightarrow B^n=I=A^{-1}{B}^2(A{)}^{m^2-1}\underset{(n-2)times}{\underset{}{B{A}^{m-1}B{A}^{m-1}\cdots B{A}^{m-1}B{A}^{m-1}A}}\Rightarrow B^n=I=A^{-1}{B}^3{A}^{(m^2-1)m}{A}^{m-1}\underset{(n-3)times}{\underset{}{B{A}^{m-1}B{A}^{m-1}\cdots B{A}^{m-1}B{A}^{m-1}A}}\Rightarrow B^n=I=A^{-1}{B}^3{A}^{(m^2-1)}\underset{(n-3)times}{\underset{}{B{A}^{m-1}B{A}^{m-1}\cdots B{A}^{m-1}B{A}^{m-1}A}}$
Similarly, we get $I=A^{-1}{B}^n(A{)}^{(m^n-1)}A$
$\Rightarrow I=A^{-1}{A}^{m^n}(\because B^n=I)\Rightarrow I=(A{)}^{(m^n-1)}\Rightarrow p=m^n-1$