How do we know that the eigenvalues of a are all real and

Question

Consider the tridiagonal matrix 

1. Why do we know that Naive Gaussian Elimination will work to solve the system of equations Ax^→ = b^→ where

for any

?

2. Solve the system of equations Ax^→ = b^→ where

using Naive Gaussian elimination

3. How do we know that the eigenvalues of A are all real and distinct before we calculate them?

4. Find the characteristic polynomial of A.

5. Find the eigenvalues of A. Hint: The characteristic polynomial can be easily factored into the product of two linear terms and a quadratic that you use the quadratic formula on.