A={{1,2},{3,4}};​​B={{5,6},{7,8}};​​U={{9,10},{11,12}};​​​​​​X=Array[x,Dimensions[A]];​​Y=Array[y,Dimensions[B]];​​​​​​equations={A.X==X.A,B.Y==Y.B,U==X.Y};​​​​​​solutions=Solve[equations,Flatten[{X,Y}]];​​solutions​​
Out[]=
{}