Question: Can all 4x4 matrices be written as a product of the matrices S and R that i've defined?

Hi! Any help would be greatly appreciated :)

I have two matrices S and R, where 

S :=

Matrix([[a_1, a_2, a_3, a_4], [b_1, b_2, b_3, b_4], [c_1, c_2, c_3, c_4], [d_1, d_2, d_3, d_4]]);

such that 

a_1*d_1 = b_1*c_1
a_2*d_2 = b_2*c_2
a_3*d_3 = b_3*c_3
a_4*d_4 = b_4*c_4;

and 

R :=

Matrix([[s_1, t_1, r_1, l_1], [s_2, t_2, r_2, l_2], [s_3, t_3, r_3, l_3], [s_4, t_4, r_4, l_4]]);

such that 

s_1*l_1 = t_1*r_1
s_2*l_2 = t_2*r_2
s_3*l_3 = t_3*r_3
s_4*l_4 = t_4*r_4.

Now I have a specific matrix T, where

T := Matrix( [ [1, 0, 0, 1], [0, 1, 0, 0], [0, 1, 0, 0], [1, 0, 0, 1] ] );

I want to know if there is a solution to T = S*R. 

Someone has already showed me that indeed

a solution to this exists (there are 90 solutions). 

But now, this might be a stretch, but I'd like to know if, 

in general, any 4x4 matrix can be written as a 

product of matrices of the form S and R 

(where S and R satisfy those constraints). 

Thanks for your help!