Linear algebra
Problem:
Let matrix
A
A be a 2x2 matrix defined as follows:
A
=
(
3
4
2
1
)
A=(3241)
Let
v
v be a vector:
v
=
(
1
3
)
v=(13)
Find the matrix-vector product
A
v
Av.
Determine if the vector
v
v is an eigenvector of the matrix
A
A, and if so, find the corresponding eigenvalue.
To find the matrix-vector product Av, we multiply the elements of each row of A by the corresponding elements of v and then sum the products:
Av = (3 * 1 + 4 * 3)
(2 * 1 + 1 * 3)
= (15)
(5)
To determine if v is an eigenvector of A, we need to check if Av is a scalar multiple of v. In other words, we need to see if there exists a scalar λ such that:
Av = λv
In this case, we have:
(15) = λ(1) (5) = λ(3)
From the second equation, we get:
λ = 5/3
Substituting this value into the first equation, we get:
15 = (5/3)(1)
This is true, so v is indeed an eigenvector of A with the corresponding eigenvalue λ = 5/3.