PROBLEM $1$bhb$.$ points $30-$ Use Python

An important problem in structural enginering is that of finding forces in a statically determinate truss $($ Fig. $1).$

This type of structure can be described by a system of coupled linear algebraic equations derived from force balances. Based on free$-$body diagrams for each node, the sum of the

forces in both horizontal and vertical directions must be zero at each node because the system is at rest.

Therefore, for node $1$:

${\displaystyle \underset{?}{\overset{?}{}}}{F}_H=0,-F_{1}cos30+F_{3}cos60+F_{1,h}=0$

${\displaystyle \underset{?}{\overset{?}{}}}{F}_V=0,-F_{1}sin30-F_{3}sin60+F_{1,v}=0$

for node $2$:

${\displaystyle \underset{?}{\overset{?}{}}}{F}_H=0F_2+H_2+F_{1}cos30+F_{2,h}=0$

${\displaystyle \underset{?}{\overset{?}{}}}{F}_V=0V_2+F_{1}sin30+F_{2,v}=0$

for node $3$:

${\displaystyle \underset{?}{\overset{?}{}}}{F}_H=0,-F_2-F_{3}cos60+F_{3,h}=0$

${\displaystyle \underset{?}{\overset{?}{}}}{F}_V=0,V_3+F_{3}sin60+F_{3,v}=0$

where $F_{i,h}$ is the external horizontal force applied to node i$,$ where positive force is to the right, and $F_{i,v}$ is the external vertical force applied to node $i,$ where positive force is upward.

Thus, in this problem, the $1000N$ downward force on node $1$ corresponds to $F_{i,v}=-1000.$

And for this problem, all other $F_{i,h}\text{'}$s and $F_{i,v}\text{'}$s are zero. Note that the directions of the internal forces and reactions are unknown. Proper applications of Newton's laws only require consistent conventions regarding direction. Also note that in this problem the forces in all members are assumed to be in tension and act to pull the adjoining nodes apart. A negative solution therefore correponds to compression. When numerical values for the external forces are substituted and the trigonometric functions evaluated, this problem reduces to a set of six linear algebraic equations with six unknowns, as above.

Your tasks:

$($points $10)$ Express this set of linear algebraic equations in matrix form

$($points $15)$ Then, use Python to solve for the forces and reactions for the case displayed

in Fig$\mathrm{.}1$

i$.$ using the inverse of the cocfickents matrix: numpy.linalg.inv

ii$.$ using directly the command: numpy.linalg.solve

ii$.$ using LU factorization: scipy.linalg.lu;

Hint: refer to sertions $9\mathrm{.}1$ and $9\mathrm{.}6$ of tertbook

iv$.$ how do the solutions of the items above compare to each otter?

$($points $5)$ Deternine the system's matrix imerse.