It's 6 six questions and need help with it
1. a. Write the following set of equations in matrix form:
8 = 6x3 + 2x2
2 - x1 = x3
5x2 + 8x1 =13
b. Multiply the matrix of coefficients by its transpose, i.e. [A][A]T
2. For the set of equations
2x2 + 5x3 = 9
2x1 + x2 + x3 = 9
3x1 + x2 = 10
a. Compute the determinant
b. Use Cramer's rule to solve for the x's.
c. Substitute your results back into the original equations to check your results.
3. Solve the following set of equations with a. Gauss elimination and b. Gauss-Jordan
x1 + x2 - x3 = -3
6x1 + 2x2 + 2x3 = 2
-3x1 + 4x2 + x3 = 1
4. Use the Gauss-Seidel method to solve the following system until the change between iterations falls below 5%
10x1 + 2x2 - x3 = 27
-3x1 -6x2 + 2x3 = -61.5
x1 + x2 + 5x3 = -21.5
5. a. Solve the following system of equations by LU decomposition
8x1 + 4x2 - x3 = 11
2x1 - x2 + 6x3 = 7
-2x1 + 5x2 + x3 = 4
b. Determine the matrix inverse. Check your results by verifying that [A][A]-1 = [I]
6. The following system of equations is designed to determine concentrations (c's in g/m3) in a series of coupled reactors as a function of the amount of mass input to each reactor (the right-hand sides in g/day).
15c1 - 3c2 - c3 = 3800
-3c1 + 18c2 - 6c3 = 1200
-4c1 - c2 + 12c3 = 2350
a. Determine the matrix inverse
b. Use the inverse to determine the solution
c. How much will the concentration in reactor 3 be reduced if the rate of mass input to reactors 1 and 2 is reduced by 500 and 250 g/day, respectively?