Denote SL2(Z) and SL3(Z)
the sets of matrices with 2 rows and 2 columns, respectively with 3 rows and 3 columns, with integer entries and their determinant equal to 1.
<spanclass=′latex−bold′>(a)</span> Let N be a positive integer and let g be a matrix with 3 rows and 3 columns, with rational entries. Suppose that for each positive divisor M of N there exists a rational number qM, a positive divisor f(M) of N and a matrix γM∈SL3(Z) such that
g=qM10001000f(M)γM10001000M.
Moreover, if q1=1, prove that det(g)=N and g has the following shape:
g=a11a21Na31a12a22Na32Na13Na23Na33,
where aij are all integers, i,j∈{1,2,3}.<spanclass=′latex−bold′>(b)</span> Provide an example of a matrix g with 2 rows and 2 columns which satisfies the following properties:
∙ For each positive divisor M of 6 there exists a rational number qM, a positive divisor f(M) of 6 and a matrix γM∈SL2(Z) such that
g=qM(100f(M))γM(100M)
and q1=1.
∙g does not have its determinant equal to 6 and is not of the shape
g=(a226a326a236a33),
where aij are all positive integers, i,j∈{2,3}.(Radu Toma)