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 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 4 :
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= 

 

= 

= 

= (4λ)[(10λ)(13 λ)+120]+
20[2(13λ)24]10[606(10λ)]
= (4λ)[13010 λ+13λ+λ²+120]+20[26+2λ24]10[6060+6λ]
= (4λ)[10+3λ+λ²]+20[2+2λ]10[6λ]
= (4λ)[λ²+3λ10]+20[2+2λ]10[6λ]
= 4λ²+12λ40λ³3λ²+10λ+40λ+4060λ
= λ³ + 1λ² + 2λ
To find roots let AλI = 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) (λ2) = 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

characteristic roots questions 4 characteristic roots questions 4 characteristic roots questions 4  
Question 2 : Determine the characteristic roots of the matrix

 
Question 3 : Determine the characteristic roots of the matrix

 
Question 5 : Determine the characteristic roots of the matrix

characteristic roots question4 
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