Matrix math is just shorthand for “regular algebra” with a lot of variables

What's pictured here is the Finite Element Method. Basically, the bridge is composed into little chunks, on which the physics solution is approximated by e.g. linear functions. Then one has to assemble and solve a system of equations (the matrix shown here) to determine the coefficients (i.e. slope) of those linear functions. Putting everything together gives a piecewise-linear approximation of the physics solution, and the smaller you make the chunks the better your solution gets. All of this is of course not done by hand, that's what computers are for.