In this page characteristic vectors of matrix 4 we are going to see how to find characteristic equation of any matrix with detailed example.
Definition :
The eigen vector can be obtained from (A λI)X = 0. Here A is the given matrix λ is a scalar,I is the unit matrix and X is the columns matrix formed by the variables a,b and c.
Another name of characteristic Vector:
Characteristic vector are also known as latent vectors or Eigen vectors of a matrix.
Question 4 :
Determine the characteristic vector of the matrix

To find eigen vector first let us find characteristic roots of the given 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
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
Substitute λ = 0 in the matrix A  λI
= 

From this matrix we are going to form three linear equations using variables x,y and z.
4x  20y  10z = 0  (1)
2x + 10y + 4z = 0  (2)
6x  30y  13z = 0  (3)
By solving (1) and (2) we get the eigen vector
The eigen vector x = 

Substitute λ = 1 in the matrix A  λI
= 

From this matrix we are going to form three linear equations using variables x,y and z.
5x  20y  10z = 0  (4)
2x + 10y + 4z = 0  (5)
6x  30y  12z = 0  (6)
By solving (4) and (5) we get the eigen vector characteristic vectors of matrix4
The eigen vector y = 

Substitute λ = 2 in the matrix A  λI
= 

From this matrix we are going to form three linear equations using variables x,y and z.
2x  20y  10z = 0  (7)
2x + 8y + 4z = 0  (8)
6x  30y  15z = 0  (9)
By solving (7) and (8) we get the eigen vector characteristic vectors of matrix 4 characteristic vectors of matrix 4 characteristic vectors of matrix 4 characteristic vectors of matrix 4
The eigen vector z = 

Questions 
Solution 
Question 1 : Determine the characteristic vector of the matrix

 
Question 2 : Determine the characteristic vector of the matrix

 
Question 3 : Determine the characteristic vector of the matrix

 
Question 5 : Determine the characteristic vector of the matrix

