In this page characteristic roots question1 we are going to see how to find characteristic roots of any given matrix.
Definition :
Let A be any square matrix of order n x n and I be a unit matrix of same order. Then AλI is called characteristic polynomial of matrix.
Then the equation AλI = 0 is called characteristic roots of matrix. The roots of this equation is called characteristic roots of matrix.
Another name of characteristic roots:
characteristic roots are also known as latent roots or eigenvalues of a matrix.
Question 1 :
Determine the characteristic roots of the matrix

Solution:
Let A = 

The order of A is 3 x 3. So the unit matrix I = 

Now we have to multiply λ with unit matrix I.
λI = 


 

= 

AλI= 
 
= (5λ)[(2λ) (5λ)  0]  0 [(0  0)] + 1 [0 (2 λ)] = (5λ)[ 10 + 2 λ  5 λ + λ²]  0 + 2 + λ = (5λ)[ 10  3 λ + λ²]  0 + 2 + λ = (5λ)[λ² 3 λ10] + 2 + λ = 5 λ²  15 λ  50  λ³ + 3 λ² + 10 λ + 2 + λ =  λ³ + 5 λ² + 3 λ²  15 λ + 10 λ + λ  50 + 2 =  λ³ + 8 λ²  4 λ  48 = λ³  8 λ² + 4 λ + 48 
To find roots let AλI = 0
λ³  8 λ² + 4 λ + 48 = 0
For solving this equation first let us do synthetic division.characteristic roots question1
By using synthetic division we have found one value of λ that is λ = 2.
Now we have to solve λ²  10 λ + 24 to get another two values. For that let us factorize
λ²  10 λ + 24 = 0
λ²  6 λ  4 λ + 24 = 0
λ (λ  6)  4 (λ  6) = 0
(λ  6) (λ  4) = 0
λ  6 = 0
λ = 6
λ  4 = 0
λ = 4
Therefore the characteristic roots (or) Eigen values are x = 2,4,6
Questions 
Solution 
Question 2 : Determine the characteristic roots of the matrix

 
Question 3 : Determine the characteristic roots of the matrix

 
Question 4 : Determine the characteristic roots of the matrix

 
Question 5 : Determine the characteristic roots of the matrix characteristic roots question 1 characteristic roots question 1

characteristic roots question 1 characteristic roots question 1 