# Practice Exam 3 M314

Note: These are problems I've used on old exams. The actual exam will probably be longer, so be sure also to study the problems on the syllabus, the quizzes, and what was covered in class. To save space, column vectors are sometimes given as the transpose of a row vector.

[1] (9 points) (a) Explain why you know, without doing any calculation at all, that the matrix A =
```[ 4  -4]
[-4  -2]
```
is diagonalizable.
(b) Find an orthogonal matrix P and a diagonal matrix D such that P-1AP = D. Also find P-1.

[2] (8 points) (a) Find an orthogonal basis for the subspace V of R3 spanned by the vectors w1 = (1,2,1) and w2 = (0,1,2).
(b) Find the projection of the vector (1,3,1) into the subspace V of part (a).

[3] (8 points) Consider the matrix A =
```[0 1 1]
[1 0 1]
[1 1 0]
```
(a) Find the eigenvalues for A. Show your work!
(b) What are the algebraic multiplicities and the geometric multiplicities of each of the eigenvalues of A?

[4] (8 points) Let A be the matrix
``` [ 3  2  6]
a[-6  b  2]
[ 2  6  c].
```
(a) Choose values for a, b and c so that A is an orthogonal matrix.
(b) Using the values of a, b and c that you chose to make A orthogonal, find A-1.