1. Given the system:

   x₁ + 2x₂ + 3x₃ = 0,

   7x₁ + 9x₂ + 8x₃ = 0 ,

   3x₁ + x₂ – 4x₃ = 0.

   1. Is the system homogeneous or nonhomogeneous?
   2. Show that   x = [0, 0, 0]^T   is a solution.
   3. Show that   x = [11, –13, 5]^T   is a solution.
   4. Show that   x = [–22, 26, –10]^T   is a solution.

2. Based on the findings in Problem 1, what is det(A)?  (Answer this without actually calculating det(A).)

3. Based on Problem 2, how many solutions does the system in Problem 1 actually have?

4. Based on Problem 2, is matrix A singular or nonsingular?

5. Use Gauss elimination/LU factorization to calculate det(A) in Problem 1.

6. What is the difference between 0 and 0 ?

7. Suppose we have a system Ax = 0.
   1. What can be said if det(A) = 0?
   2. What can be said if det(A) ≠ 0?

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