Linear Algebra
Tutorial 3
Exercise 1
Find the eigenvalues and eigenvectors of the following matrices
6
0
2
𝐴 = ( 4 −1 −1)
−3 0
1
0 −1 1
𝐵 = (−1 0 −1)
1 −1 0
2
𝐶 = (0
0
1 1
5 0)
1 5
Exercise 2
Let matrix M ∈ 𝑀5 (ℝ) with the following characteristic polynomial:
𝑃𝑀 (𝜆) = (1 − 𝜆)3 (4 − 𝜆)2
Find the eigenvalues of M
Exercise 3 (Exam 2019)
2 1
)
1 2
Consider the matrix 𝐴 = (
1. Find the eigenvalues
2. Find the eigenvectors
3. Based on the previous results, study the sign of the following quadratic form:
𝑄(𝑥1 , 𝑥2 ) = 2𝑥1 2 + 2𝑥2 2 + 2𝑥1 𝑥2
Exercise 4
0
1 −1
Let 𝐴 = ( 1
1 −1)
−1 −1 0
1. Determine the general quadratic form of A
2. Determine the sign of the quadratic form
1