The system does not appear to have a solution.

xmat = {{a[1], 1}, {b[1], -a[1]}};
cmat = {{0, 1}, {b[1] - a[1]*a[1], 0}};

In[308]:= lmat = Array[l, Dimensions[xmat]];
Solve[xmat.lmat == lmat.cmat, Flatten[lmat]]

(* Out[309]= {{l[1, 1] -> 0, l[1, 2] -> 0, l[2, 1] -> 0, l[2, 2] -> 0}} *)

General remarl: avoid capital letters for variables. Many have built-in meanings.

Okay, so same idea. I changed lmat to be explicitly unimodular lower triangular.

lmat = {{1, 0}, {z, 1}};
Solve[xmat.lmat == lmat.cmat, Variables[lmat]]

(* Out[317]= {{z -> -a[1]}} *)