Does a child object inherit the matrix from the parent? When you create a hierarchy with nesting like so:



Does Circle inherit the properties from the parent matrix? Or is first the matrix of Cirlce.001 applied followed by the matrix of Circle?

You can't tell by the values in the editor since it could subtract with the parent values.
So hopefully a programmer can answer me this.
Basics of Local and Global Coordinate systems

Each object in blender has a number of matrices. Its global transform matrix ob.matrix_world its local space matrix ob.matrix_local.  The basis matrix ob.matrix_basis, this is what you see in the "editor" 


and its parenting matrix ob.matrix_parent_inverse (This is set when choosing parenting Keep Offset for example)  

Relation between basis and local

ob.matrix_local = ob.matrix_parent_inverse @ ob.matrix_basis


As an example, select an object and calculate its global matrix by going up the hierarchy chain.

import bpy
from mathutils import Matrix

context = bpy.context
ob = context.object
M = Matrix()
print(ob.matrix_world)
while ob.parent:
    M = (ob.matrix_parent_inverse @ ob.matrix_basis) @ M
    ob = ob.parent

M = ob.matrix_basis @ M

print(M)


similarly

import bpy
from mathutils import Matrix

context = bpy.context
ob = context.object
M = Matrix()
print(ob.matrix_world)
while ob.parent:
    M = ob.matrix_local @ M
    ob = ob.parent

M = ob.matrix_local @ M

print(M)


Result on test case, Cube is parent of Cube.001 is parent of Cube.002 some random transforms applied to each.



&lt;Matrix 4x4 ( 0.3256, -0.9107,  0.8122, -3.8782)
            (-0.6125, -0.9379, -0.8588, -6.2457)
            ( 1.5886,  0.1065, -0.9282,  4.2578)
            ( 0.0000,  0.0000,  0.0000,  1.0000)&gt;
&lt;Matrix 4x4 ( 0.3256, -0.9107,  0.8122, -3.8782)
            (-0.6125, -0.9379, -0.8588, -6.2457)
            ( 1.5886,  0.1065, -0.9282,  4.2578)
            ( 0.0000,  0.0000,  0.0000,  1.0000)&gt;