4. How many elements does the multiplicative group of 2x2 matrices with elements in residue classes modulo 3 have? What about the subgroup of matrices that have a determinant of 1 modulo 3?

Supplement with exercises G.M.1/2014

NOTE: All subjects are mandatory.

Each subject is graded from 0 to 7 points.

Working time 3 hours.

Each subject should be written on a separate sheet.
4. a) Count the 81 matrices (2p)

b) Determine the number of matrices that are not invertible $(2 p)$

Put in bijection those with determinant 1 with those with determinant 2 (each representative of the respective class $(2 p)$

Obtain the $24=48: 2$ matrices with determinant $1(1 \mathrm{p})$

Note: Or list them all (5 p)