1. Given a square matrix A, to solve for eigenvalues and eigenvectors [Ax = lambda x]
a) (A-lambdaI)x=0 is equivalent, so, since we are looking for nontrivial x solutions, that means that this homogeneous system must have infinite solutions, so we can solve for det(A-lambdaI)=0.
b) Once we have a lambda that works, we can take the inverse of (A-lambdaI) to solve for the eigenvectors
c) Once we have a lambda that works, we can create the augmented matrix [A-lamdbaI|0] and reduce to solve for the nullspace of (A-lamdbaI) [called the eigenspace of A] and write out a basis.
d) a and b
e) a and c

2. A reflection matrix
a) has eigenvalue 1 on the line of reflection
b) eigenvalue -1 perpendicular to the line of reflection
c) eigenvalue -2 for some line
d) all of the above
e) more than one of a), b) c) but not all of them.

3. An eigenvector x [satisfying Ax=lambda x] allows us to turn:
a) Matrix multiplication into matrix addition
b) Matrix addition into matrix multiplication
c) Matrix multiplication into scalar multiplication
d) Matrix addition into scalar multiplication
e) none of the above

Solutions
1. e)
2. e) [a and c]
3. c)