1. (By hand) Linear Equations . (a) Write the following system of equations in the matrix form. (Ax-b form) (b) Is the coefficient matrix diagonally dominant? Why? (c) Solve it with partial pivoting 2. (By hand) Round-off error and Gaussian elimination You may still remember the impact of round-off error in root finding for an equation When solving a system of equations, the limit of available digits in computation and storage also has a great impact. Consider the following system of equations 0.7654x1 +100.1x2 3.111 234.5 159.7X2 2.456 (a) Solve using Naive Gauss Elimination using 4 significant figures (b) Use Gauss Elimination with row pivoting using 4 significant figures (c) Substitute the two sets of results back to the equation. Which method fits the original equations better? (d) Try to explore and explain what caused the difference in precision between the two methods Notes: Using 4 significant figures is same with 4-digit arithmetic with chopping. It means in every calculation step (summation, subtraction, multiplication, and division) keep only 4 digits in the result. For example, 100.0-5.102- 94.9 (because in summation and subtraction you cannot reach better precision than the input numbers), and 1.278* (100.0 - 5.102) 1.278* 94.90 121.3 3. (Programming) Gaussian elimination (a) Modify the codes from class to solve a system of equations using Gaussian Elim- ination method with pivoting. The input of the function includes the coefficient matrix A and the right hand side b. The output is the result vector x