MAT3700 Assignment 3|COMPLETE ANSWERS & SOLUTIONS| 2025

MAT3700
Assignment 3
2025

, MAT3700 — Assignment 3
(Module 5: Gauss Elimination & Eigenvalues)
QUESTION 1

3 6
Let 𝐵 = ( ). Find the eigenvalues of 𝐵. (5)
1 4

Compute characteristic polynomial det(𝐵 − 𝜆𝐼) = 0 :

3−𝜆 6 2
det ( ) = (3 − 𝜆)(4 − 𝜆) − 6 = 𝜆 − 7𝜆 + 12 − 6 = 𝜆 2 − 7𝜆 + 6.
1 4−𝜆

Solve 𝜆2 − 7𝜆 + 6 = 0 . Discriminant = 49 − 24 = 25 . So

7±5
𝜆= ⇒ 𝜆 1 = 6, 𝜆 2 = 1.
2




Eigenvalues: 𝜆 = 6 and 𝜆 = 1.




QUESTION 2

3 2 2
Let 𝐴 = ( 0 2 1) . Find an eigenvector corresponding to 𝜆 = 2 . (5)
0 0 4

Solve (𝐴 − 2𝐼)𝐯 = 𝟎 .

1 2 2
𝐴 − 2𝐼 = ( 0 0 1) .
0 0 2

Write 𝐯 = (𝑥, 𝑦, 𝑧) 𝑇 . The system is:

𝑥 + 2𝑦 + 2𝑧 = 0,
{𝑧 = 0,
2𝑧 = 0.