Eigenvalues are values where Ax = λx, where λ is the eigenvalue and x is the eigenvector. To find the eigenvalues of a matrix A, we calculate the determinant of A - λI. The values of λ where the determinant is 0 are the eigenvalues. For the given 2x2 matrix, the eigenvalues are found to be 4 and -2 by calculating the determinant of A - λI and setting it equal to 0. The corresponding eigenvectors are X = [3, 1] for the eigenvalue of 4 and X = [4, 1] for the eigenvalue of -2.

1. EigenValues and EigenVectors
EigenValues:
If x be an eigenvector of the matrix A, then there must exist an
eigenvalue X such that AX=λX.
Therefore,
λ
λ
This is the characteristic equation of A. Its roots determine the
eigenvalues of A.
Example:
Find the Eigenvalue for the given matrix.
A=
First find the det λ
for that find the λ = λ
=
λ
λ
λ
λ
λ
So that,

2. Let X=
so,
λ
Put
( )X=0
( )X=0
=0
4x1-12x2=0
x1-3x2=0
So x1=3x2 ------------------------->(a)
By using the eq(a) we get,
X=
Here c is arbitrary number.
Now λ
Put
( )X=0
( )X=0

3. =0
3x1-12x2=0
x1-4x2=0
So x1=4x2 ------------------------->(b)
By using the eq(b) we get,
Any eigenvector X of A associated to the eigenvalue -2 is given by
X=
Here c is arbitrary number.
So Eigen Vectors are X= And X=