Homework Math 231 Due on the 18th May 1 Show that the system

Homework Math 231. Due on the 18th May 1) Show that the system of equations has infinitely many solutions. Specify exactly the solutions and the limitations (Bonus: Graph the solutions) 2x2-y2+22 4x2 + y2 +z2 4 10 = = 2) 2) Use matlab or any other software of your choice to generate a 10x10 random Use matlab or any other software of your choice to generate a 10x10 random matrix A, and a 10x1 matrix B. Use the software to solve the system AX-B. 3) Use Cramer\'s rule to solve the system rn x + y + z 2x + y + z 4x-y-2z = = = A B c where A. B. C are the last digits of your ID and m is a parameter. a) Solve the system when m=1.98 and when m-2.01. Is there a big difference between the solutions? b) Solve the system when m:4 and m-4.03. Is there a big difference between the solutions? c) What can you conclude? Relate solving systems of the form AX-B to one of your previous Engineering, Stat or (Math for education )courses. Show a specific detailed application. (no computations are needed here). 4)

Solution

In linear algebra, Cramer\'s rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750,^[1] although Colin Maclaurin also published special cases of the rule in 1748^[2] (and possibly knew of it as early as 1729).^[3][4][5]

Cramer\'s rule is computationally very inefficient for systems of more than two or three equations;^[6] its asymptotic complexity is O(n·n!) compared to elimination methods that have polynomial time complexity.^[7][8] Cramer\'s rule is also numerically unstable even for 2×2 systems.^[9]