#### Please Solve R. D. Sharma class 12 Chapter 4 Algbra of Matrices Exercise Multiple Choice Questions Question  4 Maths textbook Solution.

Answer: The correct option is $\text { (B), } \mathrm{B}^{2}=\mathrm{B} \text { and } \mathrm{A}^{2}=\mathrm{A}$

Given:$A B=A, B A=B$

Solution:

$A B=A$                                                                                                         ....(1)

$BA=B$                                                                                                         ....(2)

Now from Equation (2),

\begin{aligned} &B \times(A B)=B \\ &B^{2} A=B \end{aligned}

Again from Equation (2),

\begin{aligned} &\mathrm{B}^{2} \mathrm{~A}=\mathrm{BA} \\ &\mathrm{B}^{2}=\mathrm{B} \end{aligned}

Now from Equation (1),

\begin{aligned} &A \times(B A)=A \\ &A^{2} B=A \end{aligned}

Again from Equation (1),

\begin{aligned} &\mathrm{A}^{2} \mathrm{~B}=\mathrm{AB} \\ &\mathrm{A}^{2}=\mathrm{A} \end{aligned}

So,$\mathrm{A}^{2}=\mathrm{A} \text { and } \mathrm{B}^{2}=\mathrm{B}$

Hence, the correct option is (B).