In this page characteristic vectors of matrix 2 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 2 :
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= 

 

= 

= 

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.
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
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.
3x + 1y + 3z = 0  (1)
1x + 7y + 1z = 0  (2)
3x + 1y + 3z = 0  (3)
By solving (1) and (2) we get the eigen vector characteristic vectors of matrix2
The eigen vector x = 

Substitute λ = 3 in the matrix A  λI
= 

From this matrix we are going to form three linear equations using variables x,y and z.
2x + 1y + 3z = 0  (4)
1x + 2y + 1z = 0  (5)
3x + 1y  2z = 0  (6)
By solving (4) and (5) we get the eigen vector
The eigen vector y = 

Subsitute λ = 6 in the matrix A  λI characteristic vectors of matrix2
= 

From this matrix we are going to form three linear equations using variables x,y and z.
5x + 1y + 3z = 0  (7)
1x  1y + 1z = 0  (8)
3x + 1y  5z = 0  (9)
By solving (7) and (8) we get the eigen vector
The eigen vector z = 

Questions 
Solution 
characteristic vectors of matrix 2 characteristic vectors of matrix 2
Question 1 : Determine the characteristic vector of the matrix

characteristic vectors of matrix 2  
Question 3 : Determine the characteristic vector of the matrix

 
Question 4 : Determine the characteristic vector of the matrix

 
Question 5 : Determine the characteristic vector of the matrix

characteristic vectors of matrix 2 characteristic vectors of matrix 2 
Mar 26, 23 08:27 PM
Mar 26, 23 08:26 PM
Mar 24, 23 05:25 AM