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Please Solve R.D.Sharma Class 12 Chapter 4 Algebra of Matrices Exercise MCQs Question 5 Maths Textbook Solution.

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

Given:A B=A, B A=B


\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}


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}

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

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