In this page characteristic roots question2 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 2 :
Determine the characteristic roots of the matrix

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= 

 

= 

= 

AλI= 
 
= (1λ)[ (5λ)(1λ)  1 ]  1[1  λ  3] + 3 [1  3 (5λ) ] = (1λ)[ 5  5 λ  λ + λ²  1 ]  1[ λ  2] + 3 [ 1  15 +3 λ ] = (1λ)[ λ²  6 λ + 4 ] + 1 λ + 2 + 3 [  14 +3 λ ] = λ²  6 λ + 4  λ³ + 6 λ²  4 λ + λ + 2  42 + 9 λ =  λ³ + λ² + 6 λ²  6 λ  4 λ + λ + 9 λ + 4 + 2  42 =  λ³ + 7 λ²  10 λ + 10 λ + 6  42 =  λ³ + 7 λ²  36 = λ³  7 λ² + 36 
To find roots let AλI = 0
λ³  7 λ² + 36 = 0
For solving this equation first let us do synthetic division.characteristic roots question 2 characteristic roots question 2 characteristic roots question 2 characteristic roots question 2
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  9 λ + 18 = 0
λ²  3 λ  6 λ + 18 = 0
λ (λ  3)  6 (λ  3) = 0
(λ  6) (λ  3) = 0
λ  6 = 0
λ = 6
λ  3 = 0
λ = 3
Therefore the characteristic roots (or) Eigen values are x = 2,3,6
Questions 
Solution 
Question 1 : Determine the characteristic roots of the matrix

characteristic roots question 2  
Question 3 : characteristic roots question2 Determine the characteristic roots of the matrix

 
Question 4 : Determine the characteristic roots of the matrix

 
Question 5 : characteristic roots question2 Determine the characteristic roots of the matrix


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