Characteristic Vectors of Matrix 2

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

1 1 3 1 5 1 3 1 1

To find eigen vector first let us find characteristic roots of the given matrix.

Now we have to multiply λ with unit matrix I.

To find roots let |A-λI| = 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

=	3   1   3 1   7   1 3   1   3

 

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

Substitute λ = 3 in the matrix A - λI

=	-2   1   3 1   2   1 3   1   -2

 

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

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

=	-5   1   3 1   -1   1 3   1   -5

 

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 

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• Equality of matrices
• Operation on matrices
• Algebraic properties of matrices
• Addition properties
• Multiplication properties
• Minor of matrix
• Determinant of matrix
• Adjoint of matrix
• Inverse of matrix
• Solving linear equation by inversion method
• Solving linear equations of two unknowns by cramer method
• Solving linear equations of three unknowns by cramer method
• Rank of matrix
• Solve linear equation of three unknowns by rank method
• Linear dependence of vector
• Characteristic Equation of Matrix
• Characteristic Vector of Matrix
• Diagonalization of Matrix
• Gauss Elimination Method
• Worksheet of matrices