Characteristic Roots Question 1





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 |A-λI| is called characteristic polynomial of matrix. 

Then the equation |A-λI| = 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

 
5 0 1
0 -2 0
1 0 5
 


Solution:



   Let A =

 
5 0 1
0 -2 0
1 0 5
 


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

 
5 0 1
0 -2 0
1 0 5
 

Now we have to multiply λ with unit matrix I.

  λI =

 
λ 0 0
0 λ 0
0 0 λ
 
A-λI=
 
5 0 1
0 -2 0
1 0 5
 
-
 
λ 0 0
0 λ 0
0 0 λ
 
 
                   
  =
 
(5-λ)   (0-0)   (1-0)
(0-0)   (-2-λ)   (0-0)
(1-0)   (0-0)   (5-λ)
 
 
                             
  =
 
(5-λ)   0   1
0   (-2-λ)   0
1   0   (5-λ)
 
 
A-λI=
 
(5-λ)   0   1
0   (-2-λ)   0
1   0   (5-λ)
 

  =  (5-λ)[(-2-λ) (5-λ) - 0] - 0 [(0 - 0)] + 1 [0- (-2 -λ)]

  =  (5-λ)[ -10 + 2 λ - 5 λ + λ²] - 0 + 2 + λ

  =  (5-λ)[ -10 - 3 λ + λ²] - 0 + 2 + λ

  =  (5-λ)[λ² -3 λ-10] + 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 |A-λI| = 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

 
1 1 3
1 5 1
3 1 1
 


Solution

Question 3 :

Determine the characteristic roots of the matrix

 
-2 2 -3
2 1 -6
-1 -2 0
 



Solution

Question 4 :

Determine the characteristic roots of the matrix

 
4 -20 -10
-2 10 4
6 -30 -13
 



Solution

Question 5 :

Determine the characteristic roots of the matrix  characteristic roots question 1  characteristic roots question 1

 
11 -4 -7
7 -2 -5
10 -4 -6
 



Solution

characteristic roots question 1 characteristic roots question 1







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