For the final Sage project, please use the following matrix:

(1)1. Find the determinant of this matrix. Is this matrix nonsingular?

2. Find the characteristic polynomial of this matrix. Use this polynomial to determine the eigenvalues.

3. Use Sage to find the eigenvalues and eigenvectors of this matrix.

4. Pick one eigenvector and demonstrated that it is in fact an eigenvector for the corresponding eigenvalue. (see the definition if necessary)

5. Find a matrix that has the 7 eigenvalues: 2, 3, 4, 5, 6, 7, 8

6. Optional: Use Sage to find the expanded formula for a generic $4\times 4$ matrix. How many terms are there in this formula. Do you have a guess as to how to generalize the determinant formula for $n\times n$ matrices.

Hint: The code below will generate the matrix given in the problem:

```
def CircMat(n,m):
L=[]
for i in range(n):
L.append([])
for j in range(n):
if i==j:
L[i].append(m)
else:
L[i].append(1)
return matrix(QQ,L)
```

and

`A=CircMat(10,9)`

You can change the 10 to change the size of the matrix and you can change the 9 to create a different number down the diagonals.