In this page characteristic roots questions 3 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 3 :
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= 

 

= 

= 

= (2λ)[ λ(1λ)  12 ]  2[2 λ  6]  3 [4(1)(1λ) ]
= (2λ)[ λ + λ²  12 ] + 4 λ + 12  3 [4+1λ ]
= (2λ)[ λ² λ  12 ] + 4 λ + 12  3 [3λ ]
= (2λ) [λ² λ  12 ] + 4 λ + 12 + 9 + 3 λ
= 2λ² + 2λ + 24  λ³ + λ² + 12 λ + 4 λ + 12 + 9 + 3 λ
=  λ³  λ² + 2λ + 12 λ + 4 λ + 3 λ + 24 + 12 + 9
=  λ³  λ² + 21λ + 45
= λ³ + λ²  21λ  45
To find roots let AλI = 0
λ³ + λ²  21λ  45 = 0
For solving this equation first let us do synthetic division.characteristic roots questions 3 characteristic roots questions 3
By using synthetic division we have found one value of λ that is λ = 3.
Now we have to solve λ²  2 λ  15 to get another two values. For that let us factorize
λ²  2 λ  15 = 0
λ² + 3 λ  5 λ  15 = 0
λ (λ + 3)  5 (λ + 3) = 0
(λ  5) (λ + 3) = 0
λ  5 = 0
λ = 5
λ + 3 = 0
λ =  3
Therefore the characteristic roots (or) Eigen values are x = 3,3,5
Questions 
Solution 
Question 1 : Determine the characteristic roots of the matrix

 
Question 2 : Determine the characteristic roots of the matrix

 
Question 4 : Determine the characteristic roots of the matrix

 
Question 5 : characteristic roots question3 Determine the characteristic roots of the matrix

characteristic roots questions 3 