Question 10, Exercise 2.2

Solutions of Question 10 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

If $A$ and $B$ are two matrices such that $A B=B$ and $B A=A$. Find $A^{2}+B^{2}$

Solution.

Given $$AB = B$$ and $$BA = A$$ $$\begin{align*} A^2 &= AA\\ & = A(BA)\\ &=(AB)A\\ &=BA\\ &=A \end{align*}$$ Similarly, $$\begin{align*} B^2&= BB \\ &=B(AB)\\ & = (BA)B\\ &=AB\\ &=B\end{align*}$$ Now, $$A^2 + B^2 = A + B$$ Therefore, given the conditions $AB = B$ and $BA = A$, we find that: $$A^2 + B^2 = A + B$$

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