Intro to Linear Algebra – Final Exam
with Solutions
Problem 1
Given the system below, create the augmented matrix and solve using row reduction:
x - 2x + 3x = 7₁ ₂ ₃
2x + x - x = 1₁ ₂ ₃
-3x + 4x + 2x = 10₁ ₂ ₃
Determine if the system is consistent and classify the solution.
Solution
Augmented matrix:
[1 -2 3 | 7]
[2 1 -1 | 1]
[-3 4 2 | 10]
After row reducing, we get a unique solution:
x = 1, x = 2, x = 0₁ ₂ ₃
The system is consistent with a unique solution.
Problem 2
Let a = [1; 2; -1], a = [0; -1; 2], b = [3; 0; 5]₁ ₂
(a) Is b in the span of {a , a }?₁ ₂
(b) Solve A = b, if possible.𝑥
Solution
(a) Form [A|b] = [[1 0 | 3]; [2 -1 | 0]; [-1 2 | 5]]. Row reduce: leads to inconsistency.
So, b is not in the span.
(b) No solution exists.
Problem 3
Given A = [[1 2 -1]; [2 4 -2]; [0 1 1]]:
(a) Is A = 0 linearly dependent?𝑥
(b) Find general solution.
(c) Are columns linearly independent?
Solution
(a) Yes, row reduction shows free variables.
(b) General solution: x = s[-2;1;0] + t[1;0;1]
(c) No, columns are linearly dependent.
