Question 1:
Consider the following system of equations:
3x + y - z = -2,
6x - 2y + 2z = 8,
15x - 3y + 3z = 14.
Use the Gaussian elimination method to reduce the system to upper triangular form. Carry out your calculations by hand using exact arithmetic using the methods of Subsections 1.2 and 1.3 of Unit 4.
Clearly label the operations that you use .If the set of equations has no solution, then clearly state this; if it has a unique solution ,then find it ;and if it has an infinite number of solutions, then find the general solution
Question 2:
Consider the matrix A =
|
-2 |
0 |
0 |
| A = |
-1 |
2 |
-1 |
|
1 |
5 |
-4 |
(a) Show that the eigenvalues of A are -3, -2 and 1. For each eigenvalue, find a corresponding eigenvector.
(b) Find the eigenvalues of -3A,(A-1)2 and 2A-1 + I ,justifying your answers.
(c) Using Maxima, or otherwise, apply the direct iteration procedure described in Procedure 4 on page 107 of Unit 5 to the matrix A in order to estimate the eigenvalue of largest magnitude to six decimal places. Your answer should include the computer input, together with computer output (including a plot of progress of the iteration, as in the computer session). You should annotate your answer to indicate the main features of the input and output.
(Hint: The Maxima command direct power described in the computer session for Unit 5 is helpful here.)
(d) Using Maxima, or otherwise, apply the modified inverse iteration procedure described in Procedure 6 on page 111 of Unit 5 to the matrix A with p = -1 /2 in order to try to find the eigenvalue closest to -1 /2 correct to six decimal places. Your answer should include the computer input, together with computer output (including a plot of progress of the iteration ,as in the computer session). You should annotate your answer to indicate the main features of the input and output.
Explain why the method fails to converge, and suggest a change in the parameter p that will lead to convergence to the eigenvalue of smallest magnitude.
Question 3:
Find the solution of the simultaneous linear differential equations
dx/dt =3x - 4y + 4e-t,
dy/ dt = -x + 3y - 7e-t
that satisfies the initial conditions x(0)=5, y(0)=2.
Question 4:
Consider the function
f(x,y)=x2 - xy2 + 2xy + 15x + 64.
(a) Find the stationary points of f.
(b) Classify each of the stationary points.
(c) Determine the second-order Taylor polynomial for f near (2,1).
(There is no need to expand any brackets that may appear in your answer.)
(d) The value off f when x =2 and y = -1 is 92. Suppose, however, that we know only that the value of x lies between 1.98 and 2.02,while the value of y lies between -1.01 and -0.99. Use first-order partial derivatives to investigate the accuracy of 92 as an estimate of f(x,y) for values of x and y in these ranges.