Mini-Application\tProject Equilibrium\tTemperature\tDistribution Overview A\tcommon\tproblem\tencountered\tin\tthermodynamics\tis\tthat\tof\tsolving\tfor\tthe\tequilibrium temperature\tdistribution\tof\ta\tthin\tplate\tof\tmetal. One\tway\tof\tsolving\tthis\ttype\tof\tproblem\tis\tto solve\ta\tcontinuous-time\tdifferential\tequation\tthat\tcan\tbe\tqueried\tas\ta\tfunction\tof\t(x,y)\tfor\tany continuous-valued\tposition\ton\tthe\tplate\tof\tmetal. In\tgeneral,\tthis\tsolution\tcould\tbe\texact\tgiven certain\tassumptions,\tbut\tthis\tsolution\tis\tsomewhat\tdifficult\tto\tcompute. A\tsimpler\tway\tto approximately\tsolve\tthe\tproblem\tis\tto\tdiscretize\tthe\tplate\tand\tsolve\ta\tsystem\tof\tlinear equations\t(for\texample,\tsee\tthe\trelated\ttopic\tof\t\"finite\telement\tmethods\").\tThe\tsolution\twill\tbe found\tby\tsolving\tthis\tsystem\tof\tlinear\tequations\tusing\tthe\tmatrix\tinverse. Problems Consider\tthe\tdiscretized\tsquare\tplate\tin\tFigure\t1. Figure\t1\t-\tDiscretized\tSquare\tPlate 1. Write\ta\tsystem\tof\tlinear\tequations\tfor\tthis\tsystem\tand\tclearly\tidentify\tthe\tmatrix\tA\tin\tthe matrix\tequation\tAx=b\tby\taveraging\tthe\tvalues\tat\teach\tnode\tx1,\tx2,\tx3,\tand\tx4\tby\ttaking\tthe (thermal)\taverage\tof\tits\t(four)\tadjacent\tnodes\t(with\tthe\tgiven\tboundary\tconditions). 2. Compute\tthe\tinverse\tof\tA\tusing\tan\taugmented\tmatrix\tand\trow\toperations. 3. Find\tthe\tsolution\tof\tthe\tsystem\tof\tequations\tby\tsolving\tthe\tmatrix\tequation\tAx=b\tby multiplying\tboth\tsides\t(on\tthe\tleft)\tby\tthe\tinverse\tof\tA. 4. If\tthe\tgiven\tnodes\twere\tpart\tof\ta\tlarger\t100\tby\t100\tgrid\t(with\tthis\tsquare\tin\tthe\tupper\tleft corner),\twhat\twould\tthe\tmatrix\tA\tlook\tlike\t(draw\ta\trepresentation\tof\tthis\tmatrix\tindicating zero\tvalues,\tetc.)?\t(Such\tmatrices\tare\tcalled\t\"sparse\".)