1. Compute the product Matrix([[1,2],[2,1])* Vector([1,1]) by using matrix multiplication. Next check to verify that it satisfies the definition of eigenvector: Ax =lambda x, where A=Matrix([[1,2],[2,1]) and x=Vector([1,1]).
What is lambda, the eigenvalue corresponding to x=Vector([1,1])?
a) 1
b) 2
c) 3
d) -1
e) none of the above

2. Compute the product Matrix([[1,2],[2,1])4 * Vector([1,1]) by repeatedly using the definition of eigenvector rather than matrix/vector multiplication.
Hint: A4x = A(A(A(Ax)))) via associativity, so keep substituting in Ax =lambda x and pulling out the scalar lambda. Repeate this process 4 times.

a) Vector([1,1])
b) Vector([3,3])
c) Vector([12,12])
d) Vector([81,81])
e) None of the above

3. An eigenvector 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

4. For any integer n, what will this product be? Matrix([[1,2],[2,1])n * Vector([1,1]).
a) Vector([3n,3n])
b) 3n Vector([1,1])
c) n3 Vector([1,1])
d) 3n Vector([n,n])
e) Vector([3,3])n

5. Write Vector([1,5]) as a linear combination of the eigenvectors Vector([1,1]) and Vector([1,-1])
a) 2*Vector([1,0]) + 3*Vector([0,1])
b) 2*Vector([0,1]) + 3*Vector([1,3])
c) 2*Vector([-1,1]) + 3*Vector([1,1])
d) 2*Vector([1,1]) + 3*Vector([-1,1])
e) none of the above

6. For any integer n, what will this product be? Matrix ([[1,2],[2,1]) n Vector([1,5])
a) -1 * 3n Vector([1,1]) + 3* (-2)n Vector([1,-1])
b) 3 * (-1)n Vector([1,1]) + (-2)* (3)n Vector([1,-1])
c) 3 * 3n Vector([1,1]) + (-2)* (-1)n Vector([1,-1])
d) 3 * 3n Vector([1,1]) + (-1)* (-2)n Vector([1,-1])
e) None of the above

7. How many linearly independent eigenvectors does Matrix([[1,2],[2,1]]) have?
a) 0
b) 1
c) 2
d) infinite
e) none of the above

8. How many eigenvectors does Matrix([[1,2],[2,1]]) have?
a) 0
b) 1
c) 2
d) infinite
e) none of the above