Question

If two matrices $A$ and $B$ are such that they follow commutative property, then $A^4{B}^2$ is equal to

• A. $B^2{A}^4$
• B. $B{A}^{3}B$
• C. $A^3{B}^{2}A$
• D. $A{B}^{3}A$

Answer 1

Answer: (A), (C)

$A^3{B}^{2}A$

Since, matrices $A$ and $B$ follow commutative property.
$\Rightarrow AB=BA$ and $A{B}^n=B^{n}A$
Now,
$A^4{B}^2=A^3\left(AB\right)B=A^3\left(BA\right)B=A^{3}B\left(AB\right)=A^{3}B\left(BA\right)=A^3{B}^{2}A$
and $B^2{A}^4=B\left(B{A}^4\right)=B\left(A^{4}B\right)=\left(B{A}^4\right)B=A^{4}BB=A^4{B}^2,$
and $B{A}^{3}B=\left(A^{3}B\right)B=A^3{B}^2=A^{4}B$
and
$A{B}^{3}A=A\left(A{B}^3\right)=A^2{B}^3$
Note : when two matrices $A$ and $B$ commute , degree of matrices in $A^m{B}^n$ remains same .