Question

Consider the following statement :
$A:\text{If}AB=A\text{and}BA=B\text{then}{A}^n+B^n=A+B$
$R:A\text{and}B$ are idempotent matrices.
Then the correct option is

• A. Both statements $A$ and $R$ are True and $R$ is correct explanation of $A.$
• B. Both $A$ and $R$ are True but $R$ is not correct explanation to $A$
• C. $A$ is true $R$ is False
• D. $A$ is false $R$ is true

Answer 1

The correct option is A Both statements $A$ and $R$ are True and $R$ is correct explanation of $A.$

Reason
$A^2=A.A=AB.A$
$=A.B.A$
$=A.B$
$=A$

Similarly, $B^2=B$
Hence, $A$ and $B$ are idempotent.
$\Rightarrow A^n+B^n=A+B$
So, Both statements are true and $R$ is correct explanation to $A.$