Question

(a) Are there matrices $A,B\in M_n\left(R\right)$ such that $AB-BA=I$.

(b) Suppose that $A,B\in M_n\left(R\right)$ such that $(AB-BA{)}^2=AB-BA$. Show that $A$ and $B$ are commutable.

Answer 1

Solution

(a) Are there matrices $A,B\in M_n\left(R\right)$ such that $AB-BA=I$. we have$$\begin{array}{cc}AB-BA& =I\\ A^{2}B-B{A}^2& =I\\ A^2{B}^2-B^2{A}^2& =AB\\ AB(I-AB)& =B^2{A}^2\\ AB(AB-BA-AB)& =B^2{A}^2\\ AB(-BA)& =B^2{A}^2\end{array}$$Therefore, $A,B\in M_n\left(R\right)$ such that $AB-BA=I■$ 

(b) Suppose that $A,B\in M_n\left(R\right)$ such that $(AB-BA{)}^2=AB-BA$. Show that $A$ and \par $B$ are commutable. we have $\iff (AB-BA)(AB-BA-I)=0$ From $\left(a\right)\Rightarrow AB-BA=0\Rightarrow AB=BA$ Then$$\begin{array}{cc}AB-BA& =(AB-BA)(AB-BA)\\ & =ABBA-ABBA-BAAB+BABA\\ & =(AB{)}^2-A{B}^{2}A-B{A}^{2}B+(BA{)}^2\end{array}$$Therfore, $A$ and $B$ are commutable $(AB-BA{)}^2=AB-BA■$