Match the following

	List-I		List-II
A	if A is idempotent then $(I-A{)}^n$ is ------	p	$2^{n-1}(I-A)$
B	if A is involuntary then$(I-A{)}^n$  is ------	q	$I-nA$
C	if A is nilpotent of index 2 then $(I-A{)}^n$ is ------	r	O
D	If $A, B, A + I, A + B$ are idempotent matrices then AB + BA is equal to	s	I – A

a) Since A is idempotent, hence $A^2=A$

$\Rightarrow A^3=A{A}^2=AA=A^2=A,A^4=A{A}^3=AA=A^2=A\Rightarrow A^n=A$

$\Rightarrow {(I-A)}^n{=}^n{C}_{0}I{-}^n{C}_{1}A{+}^n{C}_2{A}^2{-}^n{C}_3{A}^3+\mathrm{....}$

$=$$I+({-}^n{C}_1{+}^n{C}_2{-}^n{C}_3+\mathrm{....})A-$$I[{(}^n{C}_0{-}^n{C}_1{+}^n{C}_2{-}^n{C}_3+\mathrm{....}){-}^n{C}_0]A=I-A$

b) A is in involunatary. Hence $A^2=I\Rightarrow A^3=A^5=\mathrm{....}=A\text{\hspace{0.17em}and\hspace{0.17em}}{A}^2=A^4=A^6=\mathrm{...}=I$

$\Rightarrow {(I-A)}^n{=}^n{C}_{0}I{-}^n{C}_{1}A{+}^n{C}_2{A}^2{-}^n{C}_3{A}^3+\mathrm{.....}$

$={(}^n{C}_0{+}^n{C}_2{+}^n{C}_4+\mathrm{...})I-{(}^n{C}_{1}A{+}^n{C}_3{+}^n{C}_5+\mathrm{....})A=2^{n-1}(I-A)$

c) If is nilpotent of index 2, then $A^2=A^3=A^4\mathrm{...}=A^n=0$

$\Rightarrow {(I-A)}^n{=}^n{C}_{0}I{-}^n{C}_{1}A{+}^n{C}_2{A}^2{-}^n{C}_3{A}^3+\mathrm{....}$

$=I-nA+O+O+\mathrm{...}=I-nA$

d) Given A, B, A + I, A + B are idempotent. Hence

$A^2=A,B=B^2,{(A+I)}^2=A+I$ and ${(A+B)}^2=A+B$

$\Rightarrow A^2+B^2+AB+BA=A+B\Rightarrow A+B+AB+BA=A+B$

$\Rightarrow AB+BA=O$