Question 161620
I am going to use I instead of In for the the identity. So we are given that A^3 - 7A^2 + 5I = 0 therefore A^3 = 7A^2 - 5I.

We are interested in A^4 which is A*A^3, but we know what A^3 is, so substitute A^4 = A(7A^2-5I) = 7A^3-5A. WE now have another A^3 we can substitute for, so do that. A^4 = 7(7A^2-5I)-5A = 49A^2-5A-35I.