Characteristic Roots Questions 4





In this page characteristic roots questions 4 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 4 :

Determine the characteristic roots of the matrix

 
4 -20 -10
-2 10 4
6 -30 -13
 




   Let A =

 
4 -20 -10
-2 10 4
6 -30 -13
 

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=
 
4 -20 -10
-2 10 4
6 -30 -13
 
-
 
λ 0 0
0 λ 0
0 0 λ
 
 
                      
  =
 
(-4-λ)   (-20-0)   (-10-0)
(-2-0)   (10-λ)   (4-0)
(6-0)   (-30-0)   (-13-λ)
 
 
  =
 
(-4-λ)   -20   -10
-2   (10-λ)   4
6   -30   (-13-λ)
 
 

= (4-λ)[(10-λ)(-13- λ)+120]+

    20[-2(-13-λ)-24]-10[60-6(10-λ)]

= (4-λ)[-130-10 λ+13λ+λ²+120]+20[26+2λ-24]-10[60-60+6λ]

= (4-λ)[-10+3λ+λ²]+20[2+2λ]-10[6λ]

= (4-λ)[λ²+3λ-10]+20[2+2λ]-10[6λ]

= 4λ²+12λ-40-λ³-3λ²+10λ+40λ+40-60λ

= -λ³ + 1λ² + 2λ

To find roots let |A-λI| = 0

   -λ³ + 1λ² + 2λ = 0

For solving this equation -λ from all the terms characteristic roots question4

-λ (λ² - 1λ - 2) = 0

-λ = 0 (or) λ² - 1 λ - 2 = 0

λ = 0       (λ+1) (λ-2) = 0 

               λ + 1 = 0       λ - 2 = 0

                  λ = - 1            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

characteristic roots questions 4  characteristic roots questions 4 characteristic roots questions 4

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 5 :

Determine the characteristic roots of the matrix

 
11 -4 -7
7 -2 -5
10 -4 -6
 



Solution

characteristic roots question4



“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are: 

It subtracts sadness and adds happiness in our life.    

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”






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