Question

If two matrices $A$ and $B$ are commutative, then which of the following is(are) not equal to $A^2{B}^5$

• A. $B^3{A}^5$
• B. $B{A}^5$
• C. $A{B}^{4}A$
• D. $A{B}^{5}A$

Answer 1

Answer: (A), (B), (C)

$A{B}^{4}A$

Since, matrices $A$ and $B$ are commutative
$\Rightarrow AB=BA$ and $A{B}^n=B^{n}A$
Now,
$A^2{B}^5=A\left(AB\right)B^4=A\left(BA\right)B^4=AB\left(AB\right)B^3=A{B}^2\left(AB\right)B^2=A{B}^3\left(AB\right)B=A{B}^4\left(AB\right)=A{B}^{4}BA=A{B}^{5}A$
Note : when two matrices $A$ and $B$ commute , degree of matrices in $A^m{B}^n$ remains same .