Mathematics → Subject Test → Eigenvalues and Eigenvectors
Suppose A is a square matrix. The relation for finding Eigenvalue λ corresponds to the Eigenvector X is
AX = λX
Practice Problems
Example 1. Find Eigenvalue corresponds to the Eigenvector
for matrix
Solution
As we know that AX = λX
So,
Hence, λ = 7 Answer
How to find Eigenvalues of a matrix?
To find eigenvalues of a given matrix, we have to use the relation
det ( λI – A ) = 0
Example 2. Find all the eigenvalues of the matrix
Solution
As we know that det ( λI – A ) = 0
Now,
Now find determinant of λI – A
det ( λI – A ) = ( λ–4 ) ( λ–11 ) – ( –3 ) ( 4 )
= λ2 – λ11 – λ4 + 44 + 12
= λ2 – λ15 + 56
Now set this equal to zero to obtain det ( λI – A ) = 0 and solve for the eigenvalues.
λ2 – 15λ + 56 = 0
λ2 – 8λ – 7λ + 56 = 0
λ( λ – 8 ) – 7 ( λ – 8 ) = 0
( λ – 7 ) ( λ – 8 ) = 0 ⇒ λ1 = 7, and λ2 = 8
So we get two eigenvalues. Answer