Matrix double modulo multiplication to get identity

Question

I have to multiply to matrices A and B which can consist of numbers 0,2,3,4,5,6 to get an identity matrix, however multiplication happens with moduli after every step. e.g.:

[A1 A2 A3]    and      [B1 B2 B3]
[A4 A5 A6]             [B4 B5 B6]
[A7 A8 A9]             [B7 B8 B9]

((A1*B1)%7+(A2*B4)%7+(A3*B7)%7)%7 = 1 

Which would be the element I_11

How can i find two matrices A and B?

Answer 1

$$\pmatrix{2&0&0\cr0&2&0\cr0&0&2\cr}{\rm\ and\ }\pmatrix{4&0&0\cr0&4&0\cr0&0&4\cr}$$ These two matrices use only numbers from the available set, and, modulo 7, their product is the identity matrix.