Assume initially that all the numbers are different and that, since there are 9 of them, they represent the integers from 1 to 9. Call the unknown known numbers A to I, in order:
A-B+C=1
D-E*F=-1
G/H*I=16
A+D-G=4
B+E/H=12
C+F-I=0
GI=16H; if H=1, (G,I)=(2,8) or (8,2); if H=2 (G,I)=(4,8) or (8,4); if H=3 (G,I)=(6,8) or (8,6) so H is 1, 2 or 3. But I=C+F, so if H=1, (G,C,F)=(2,5,3), (2,3,5) and I=8 since I=2 is not possible. So {A B D E}={4 6 7 9}. But if G=2, A+D=6, and no values fit because there are no numbers in the set summing to 6. Therefore H must be 2 or 3.
If H=2, G=8 and I=4, (C,F)=(1,3) or (3,1) and (A,D)=(5,7) or (7,5) leaving {B E}={6 9). From this we have E=6 and B=9, so that B+E/H=12. What about the other equations? A-B+C=1: since B=9, A+C=10 so A=7 and C=3, leaving F=1 and D=5. So the solution so far gives us (A,B,C,D,E,F,G,H,I)=(7,9,3,5,6,1,8,2,4). Substitute these in the equations:
7-9+3=1 OK
5-6*1=-1 OK
8/2*4=16 OK
7+5-8=4 OK
9+6/2=12 OK
3+1-4=0 OK
So we have a solution! Number matrix is:
7 9 3
5 6 1
8 2 4