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 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 5 :
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= 
 
= (11λ)[(2λ)(6λ)20]+4[7(6λ)+50]7[2810(2λ)] = (11λ)[(2+λ)(6+λ)20]+4[427λ+50]7[28+20+10λ] = (11λ)[12+2λ+6λ+λ²20]+4[87λ]7[8+10λ] = (11λ)[λ²+8λ8]+3228λ+5670λ = 11λ²+8λ88λ³8λ²+88λ98λ+88 =  λ³+3λ²2λ = λ³+3λ²2λ = λ(λ²3λ²+2) 
To find roots let AλI = 0
λ(λ²3λ²+2) = 0
λ = 0
Now we have to solve λ²3λ²+2 to get another two values. For that let us factorize
λ²3λ²+2 = 0
λ²1λ2λ+2 = 0
λ(λ1)2(λ1) = 0
(λ1)(λ2) = 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

 
Question 2 : Determine the characteristic roots of the matrix

 
Question 3 : Determine the characteristic roots of the matrix

 
Question 4 : Determine the characteristic roots of the matrix

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