4. (Applying LU and LUP decompositions, 14pt) In this problem, we'll use the LU/LUP decomposition to solve a linear system of equations. a) [3pt] For A=[4141147;74283;1010889;1196212;610102013]; find matrices P, L, and U so that PA = LU using Matlab's lu function. Based on your results: did Matlab use pivoting during the lu-computation? b) [4pt] For b=[4;24;7;16;6] solve Ax=b using the LU decomposition as follows. Solving Ax=b is the same as solving PAx =Pb. (With P from a). Since PA =LU, we need to solve LUx =Pb, and we can split that into two triangular systems as follows: Ly=Pb, and Ux=y Solve both of these systems using Matlab's linsolve, and state x and y explicitly. Hint: which one do you need to solve first? Remember what you have, and what you need. c) [2 pt] Compare the quality of the x you found in b to the solution of Ax=b you get from using linsolve by comparing the differences Ax - b. d) [1pt] You want so solve Ax=b for more than one vector b, so you collect the b-vectors into a single matrix B. Your goal then is to find a matrix X so that AX=B (one column x in X for each column b in B). (Check: If A is an mxn matrix, and B is a kxp matrix, what are the dimensions of X ?) For 23] solve AX=B using linsolve, and check that AX=B in Matlab. Hint: linsolve(A,B) works. e) [4pt] We now want to solve AX=B (with A from part a, and B from part d) using the LUP decomposition from parts a/b. Solving AX=B is the same as solving PAX=PB, so, as earlier, we can split this into two triangular systems: LY=PB and UX=Y. Solve these two equations using Matlab's linsolve. First find Y in LY =PB, and then use that to find X in UX=Y. Check that AXB is close to the zero matrix. For the last two problems, be sure to work with format short (or even format compact) so that the matrices don't use up too much screenspace