Consider the two-dimensional "house" constructed out of bars, as in the accompanying picture. The bottom nodes are fixed. The width of the house is 3 units, the height of the vertical sides 1 unit, and the peak is 1.5 units above the base.

(a) Determine the reduced incidence matrix A for this structure.
(b) How many distinct modes of instability are there? Describe them geometrically, and indicate whether they are mechanisms or rigid motions.
(c) Suppose we apply a combination of forces to each non-fixed node in the structure. Determine conditions such that the structure can support the forces. Write down an explicit nonzero set of external forces that satisfy these conditions, and compute the corresponding elongations of the individual bars. Which bar is under the most stress?
(d) Add in a minimal number of bars so that the resulting structure can support any force. Before starting, decide, from general principles, how many bars you need to add.
(e) With your new stable configuration, use the same force as before, and recomputed the forces on the individual bars. Which bar now has the most stress? How much have you reduced the maximal stress in your reinforced building?