Characteristic vectors of matrix :
Here we are going to see how to find characteristic equation of any matrix with example problems.
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 names of characteristic Vectors of matrix :
Characteristic vector are also known as latent vectors or Eigen vectors of a matrix.
Example :
Determine the characteristic vector of the matrix

Solution :
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(1λ)  3] + 3 [1  3 (5λ)] = (1λ) [55λλ+λ²1]  1 [1 λ  3] + 3[115+3λ] = (1λ) [λ²6λ+4]  1[2  λ] + 3[14+3λ] = λ²  6λ + 4  λ³ + 6λ²  4λ + 2 + λ  42 + 9λ =  λ³ + 7 λ²  10λ + 10λ + 6  42 =  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 = 3.
Now we have to solve λ²  4 λ  12 to get another two values.For that let us use factoring method.characteristic vectors of matrix
λ²  4 λ  12 = 0
λ²  6 λ + 2 λ  12 = 0
λ (λ6) + 2 (λ6) = 0
(λ+2) (λ6) = 0
λ + 2 = 0 λ  6 = 0
λ = 2 λ = 6
Therefore the characteristic roots are x = 3,2 and 6
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  (1)
1x + 2y + 1z = 0  (2)
3x + 1y  2z = 0  (3)
By solving (1) and (2) we get the eigen vector
The eigen vector x = 

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  (4)
1x + 7y + 1z = 0  (5)
3x + 1y + 3z = 0  (6)
By solving (4) and (5) we get the eigen vector
The eigen vector y 

Substitute λ = 6 in the matrix A  λI
= 

The eigen vector z 

Let P = 

The column of P are linearly independent eigen vectors of A . Therefore the diagonal matrix = 

(1) Determine the characteristic vector of the matrix

(2) Determine the characteristic vector of the matrix

(3) Determine the characteristic vector of the matrix

(4) Determine the characteristic vector of the matrix

(5) Determine the characteristic vector of the matrix

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