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

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- 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.

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