Characteristic Roots Questions 5





In this page characteristic roots questions 5 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 5 :

Determine the characteristic roots of the matrix

 
11 -4 -7
7 -2 -5
10 -4 -6
 




   Let A =

 
11 -4 -7
7 -2 -5
10 -4 -6
 

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

 
1 0 0
0 1 0
0 0 1
 

Now we have to multiply λ with unit matrix I.

  λI =

 
λ 0 0
0 λ 0
0 0 λ
 
A-λI=
 
11 -4 -7
7 -2 -5
10 -4 -6
 
-
 
λ 0 0
0 λ 0
0 0 λ
 
 
                      
  =
 
(11-λ)   (-4-0)   (-7-0)
(7-0)   (-2-λ)   (-5-0)
(10-0)   (-4-0)   (-6-λ)
 
 
  =
 
(11-λ)   -4   -7
7   (-2-λ)   -5
10   -4   (-6-λ)
 
 
A-λI=
 
(11-λ)   -4   -7
7   (-2-λ)   -5
10   -4   (-6-λ)
 

  =  (11-λ)[(-2-λ)(-6-λ)-20]+4[7(-6-λ)+50]-7[-28-10(-2-λ)]

  =  (11-λ)[(2+λ)(6+λ)-20]+4[-42-7λ+50]-7[-28+20+10λ]

  =  (11-λ)[12+2λ+6λ+λ²-20]+4[8-7λ]-7[-8+10λ]

  =  (11-λ)[λ²+8λ-8]+32-28λ+56-70λ

  =  11λ²+8λ-88-λ³-8λ²+88λ-98λ+88

  =  - λ³+3λ²-2λ

  =  -λ³+3λ²-2λ

  =  -λ(λ²-3λ²+2)

To find roots let |A-λI| = 0

 -λ(λ²-3λ²+2) = 0

λ = 0

Now we have to solve λ²-3λ²+2 to get another two values. For that let us factorize 

   λ²-3λ²+2 = 0

λ²-1λ-2λ+2 = 0

λ(λ-1)-2(λ-1) = 0

(λ-1)(λ-2) = 0

 λ - 1 = 0

 λ = 1

 λ - 2 = 0

 λ = 2

Therefore the characteristic roots (or) Eigen values are x = 0,1,2


Questions

Solution


Question 1 :

Determine the characteristic roots of the matrix

 
5 0 1
0 -2 0
1 0 5
 



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

characteristic roots questions 5 characteristic roots questions 5







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