MAT3700
ASSIGNMENT 3 2025
UNIQUE NO.
DUE DATE: 2025
, QUESTION 1 Given:
3 6
𝐵= ቂ ቃ
1 4
We find eigenvalues by solving:
det(𝐵 − 𝜆𝐼) = 0
3−𝜆 6
det ቂ ቃ= 0
1 4−𝜆
(3 − 𝜆)(4 − 𝜆) − (6)(1) = 0
12 − 3𝜆 − 4𝜆 + 𝜆 2 − 6 = 0
𝜆2 − 7𝜆 + 6 = 0
Factor:
(𝜆 − 6)(𝜆 − 1) = 0
Eigenvalues: 𝜆1 = 6, 𝜆2 = 1
QUESTION 2 Given:
3 2 2
𝐴 = 0 2 1൩, 𝜆=2
0 0 4
We solve:
(𝐴 − 2𝐼)𝑥 = 0
1 2 2
𝐴 − 2𝐼 = 0 0 1൩
0 0 2
From row 2: 0𝑥1 + 0𝑥2 + 1𝑥3 = 0 ⇒ 𝑥 3 = 0 From row 1: 𝑥1 + 2𝑥2 + 2(0) = 0 ⇒ 𝑥 1 =
−2𝑥2 𝑥2 is free. Let 𝑥2 = 𝑡, then 𝑥1 = −2𝑡, 𝑥3 = 0.
ASSIGNMENT 3 2025
UNIQUE NO.
DUE DATE: 2025
, QUESTION 1 Given:
3 6
𝐵= ቂ ቃ
1 4
We find eigenvalues by solving:
det(𝐵 − 𝜆𝐼) = 0
3−𝜆 6
det ቂ ቃ= 0
1 4−𝜆
(3 − 𝜆)(4 − 𝜆) − (6)(1) = 0
12 − 3𝜆 − 4𝜆 + 𝜆 2 − 6 = 0
𝜆2 − 7𝜆 + 6 = 0
Factor:
(𝜆 − 6)(𝜆 − 1) = 0
Eigenvalues: 𝜆1 = 6, 𝜆2 = 1
QUESTION 2 Given:
3 2 2
𝐴 = 0 2 1൩, 𝜆=2
0 0 4
We solve:
(𝐴 − 2𝐼)𝑥 = 0
1 2 2
𝐴 − 2𝐼 = 0 0 1൩
0 0 2
From row 2: 0𝑥1 + 0𝑥2 + 1𝑥3 = 0 ⇒ 𝑥 3 = 0 From row 1: 𝑥1 + 2𝑥2 + 2(0) = 0 ⇒ 𝑥 1 =
−2𝑥2 𝑥2 is free. Let 𝑥2 = 𝑡, then 𝑥1 = −2𝑡, 𝑥3 = 0.
