Elementary linear algebra, matrix and inverse matrix?

Question

Let
B=
1 2
5 3

If (5I - 8A)^-1 = B
find A

Answer 1

5I -8A =
5 -8a .. -8b
-8c ..... 5 - 8d
(5I -8A)^-1=
5-8d .. 8b
8c ..... 5-8a
divided by: ( 5-8a)(5-8d) -64bc
so you get a set of equations:
5-8d = ( 5-8a)(5-8d) -64bc
8b = 2[( 5-8a)(5-8d) -64bc ]
8c = 5 [( 5-8a)(5-8d) -64bc ]
5-8a = 3 [( 5-8a)(5-8d) -64bc ]
good luck!

Answer 2

5I - 8A = B^(-1)

A = 1/8 (5I - B^(-1)).

I.e., the problem is trivial except for the step of actually inverting B.

And that step I'll leave to you. ;)