#### Please Solve R.D.Sharma Class 12 Chapter 4 Algebra of Matrices Exercise Multiple Choice Questions Question 5 Maths Textbook Solution.

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

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

Solution:

$\mathrm{AB}=\mathrm{A}$                                                                                                                    .....(1)

$\mathrm{BA}=\mathrm{B}$                                                                                                                    .....(2)

Now from Equation (2),

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

Again from Equation (2),

$\mathrm{B}^{2} \mathrm{~A}=\mathrm{BA}$

$\mathrm{B}^{2}=\mathrm{B}$

Now from Equation (1),

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

Again from Equation (1),

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

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

Thus,$\mathrm{A}^{2}+\mathrm{B}^{2}=\mathrm{A}+\mathrm{B}$

So, the option (C) is the correct option.