Given a finite field F, let M2(F) denote the set of all 2 × 2 matrices with entries from F. As in Example 14.2, (M2(F), +, €¢) becomes a noncommutative ring with unity.
a) Determine the number of elements in M2(F) if F is
i) Z2 ii) Z3 iii) Zp, pa. prime
(b) As in Exercise 13 of Section 14.1, A =

ˆˆ M2(Zp) is a unit if and only if ad - be ‰  z. This occurs if the first row of A does not contain all zeros (that is, z's) and the second row is not a multiple (by an element of Zp) of the first. Use this observation to determine the number of units in
i) M2(Z2) ii) M2(Z3) iii) M2(ZP), p a prime

a c d