Question

Say true or false:

Let A, B be two matrices, such that $AB=A$ and $BA=B$, then $A^2=A$ and $B^2=B$.

• A. True
• B. False

Answer 1

The correct option is A True

$AB=A\Rightarrow B\left(AB\right)=B\left(A\right)\Rightarrow BAB=BA\Rightarrow BB=B\Rightarrow B^2=BBA=B\Rightarrow A\left(BA\right)=A\left(B\right)\Rightarrow ABA=AB\Rightarrow AA=A\Rightarrow A^2=A$