Instructions: Show all of your work; your work must justify your answers.

[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

[2] (8 points) (a) Find an orthogonal basis for the subspace V of

(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