JEE Main 2023 · 24 January · Morning Shift · Mathematics · Matrices
D. A2B=BA2{A}^{2}B = B{A}^{2}A2B=BA2
Given:
A2B=A2+B{A}^{2}B = {A}^{2} + BA2B=A2+B
⇒A2B−A2−B+I=I⇒ {A}^{2}B - {A}^{2} - B + I = I⇒A2B−A2−B+I=I
⇒A2(B−I)−(B−I)=I⇒ {A}^{2}\left(B - I\right) - \left(B - I\right) = I⇒A2(B−I)−(B−I)=I
⇒(A2−I)(B−I)=I⇒ \left({A}^{2} - I\right)\left(B - I\right) = I⇒(A2−I)(B−I)=I
⇒(A2−I)(B−I)(B−I)−1=I(B−I)−1⇒ \left({A}^{2} - I\right)\left(B - I\right){\left(B - I\right)}^{- 1} = I{\left(B - I\right)}^{- 1}⇒(A2−I)(B−I)(B−I)−1=I(B−I)−1
⇒(A2−I)=(B−I)−1⇒ \left({A}^{2} - I\right) = {\left(B - I\right)}^{- 1}⇒(A2−I)=(B−I)−1
Also,
A2+B=A2B{A}^{2} + B = {A}^{2}BA2+B=A2B
⇒A2(B−I)=B⇒ {A}^{2}\left(B - I\right) = B⇒A2(B−I)=B
⇒A2=B(B−I)−1⇒ {A}^{2} = B{\left(B - I\right)}^{- 1}⇒A2=B(B−I)−1
⇒A2=B(A2−I)⇒ {A}^{2} = B\left({A}^{2} - I\right)⇒A2=B(A2−I)
⇒A2=BA2−B⇒ {A}^{2} = B{A}^{2} - B⇒A2=BA2−B
⇒A2+B=BA2⇒ {A}^{2} + B = B{A}^{2}⇒A2+B=BA2
⇒A2B=BA2⇒ {A}^{2}B = B{A}^{2}⇒A2B=BA2
