hint:

suppose A is a matrix such that (I + E)A = I.

then A + EA = I.

applying E to both sides we have:

E(A + EA) = E

EA + E^{2}A = E

but E^{2}= E (since E is a projection) thus:

2EA = E, or EA = (1/2)E.

now substitute this back in the equation

A + EA = I, and solve for A.