CS3331: Numerical Methods
Quiz 1
March. 21, 1997

1.

(5%) Evaluate the determinant of A^-1 where

A = BCD,

and

$$B = \left[ \begin{array} {rrr} 1 & 2 & 1 \\ 0 & 2 & 3 \\ 0 &... ...rr} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 5 & 2 & 5 \\ \end{array}\right].$$

2.

(5%) Find the eigenvalues and corresponding eigenvectors of the matrix A:

$$A = \left[ \begin{array} {rr} cos(2\theta) & sin(2\theta) \\ sin(2\theta) & -cos(2\theta)\\ \end{array}\right]$$

(You should make the eigenvectors unit-length and simplify them as much as possible.)

3.

(10%) Solve the following equations

$$\left\{ \begin{array} {rrr} 2x_1 + 8 x_2 + 16 x_3 & = & 10... ... & = & 10\\ 4x_1 + 8 x_2 + 8 x_3 & = & 4\\ \end{array}\right.$$

by Gauss elimination with pivoting. You should start with the matrix

$$\left[ \begin{array} {rrrr} 2 & 8 & 16 & 10\\ 3 & 8 & 15 & 10\\ 4 & 8 & 8 & 4\\ \end{array}\right]$$

and proceed the elimination step-by-step.

4.

(10%) Use Gauss elimination to find the LU decomposition of the matrix A step-by-step:

$$A = \left[ \begin{array} {rrr} 1 & 1 & 2\\ 2 & 1 & 3\\ 3 & 3 & 7\\ \end{array}\right]$$

No pivoting is necessary.