Firstly I apologise for the use of an image but I am still unfamiliar with using the math text on here.
So I am currently studying linear algebra at second year level and have been working on this question.
I have proved a) by squaring the matrix and obtaining that it is equal to the original matrix.
I have also shown b) by saying that
$$(I-B)^2= I^2 - 2IB + B^2$$
Which then equates to $I - B$ due to the fact that $I^2=I$ and $B^2=B$ for Idempotency.
I have also shown that for c) the determinants may only equal 0 or 1 by saying:
$det(C^2)= det(C)$
$det(c)* (det(c) - 1)=0$
However I'm now stuck on how to approach questions d) and e).
Any help would be greatly appreciated.
I'll expand my comments as required.
For (d): If $D$ is the matrix of $S$ and $I_n-D$ of $S'$, then the matrix of $S\circ S'$ is $$D(I_n-D)=D-D^2=D-D=0$$ so $S\circ S'$ is the null aplication.
For (e): As the matrix of $T$, $E$, is idempotent, then $E^2=E$. Also the matrix of $T^{9001}$ is $$E^{9001}=E^2 \cdot E^{8999}=E\cdot E^{8999}=E^{9000}=\ldots=E$$ so $T^{9001}=T$.