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# If $A,B$ are two square matrices such that $AB=A$ and $BA=B$ then $A$ and $B$ are

$\begin{array}{1 1}(A)\;\text{nilpotent}&(B)\;\text{idempotent}\\(C)\;\text{periodic}&(D)\;\text{involutory}\end{array}$

Toolbox:
• In algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself.[1][2] That is, the matrix M is idempotent if and only if MM = M. For this product MM to be defined, M must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings.
Given :
$AB=A$
$BA=B$
Now $AB=A$-----(1)
Multiplying (1) by $A$, $\rightarrow (AB)A=A^2$
$A(BA)=A^2$(By associative law)
$BA=B$
$\Rightarrow AB=A^2$-------(2)
From eq(1) & eq(2) we get
$A^2=A$
$B^2=B$
$\therefore$ A and B are idempotent matrices.
Hence (B) is the correct answer.
edited Jul 11, 2014