Characteristic Vectors of Matrix 5

In this page characteristic vectors of matrix 5 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 5 :

Determine the characteristic vector of the matrix

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

 -λ(λ²-3λ²+2) = 0

λ = 0

Now we have to solve λ²-3λ²+2 to get another two values. For that let us factorize 

   λ²-3λ²+2 = 0

λ²-1λ-2λ+2 = 0

λ(λ-1)-2(λ-1) = 0

(λ-1)(λ-2) = 0

 λ - 1 = 0

 λ = 1

 λ - 2 = 0

 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.

11x - 4y - 7z = 0  ------ (1)

7x - 2y - 5z = 0  ------ (2)

10x - 4y - 6z = 0  ------ (3)

By solving (1) and (2) we get the eigen vector   characteristic vectors of matrix 5  characteristic vectors of matrix 5  characteristic vectors of matrix 5

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.

10x - 4y - 7z = 0  ------ (4)

7x - 3y - 5z = 0  ------ (5)

10x - 4y - 7z = 0  ------ (6)

By solving (4) and (5) we get the eigen vector

Substitute λ = 2 in the matrix A - λI       characteristic vectors of matrix5

From this matrix we are going to form three linear equations using variables x,y and z.

9x - 4y - 7z = 0  ------ (7)

7x - 4y - 5z = 0  ------ (8)

10x - 4y - 8z = 0  ------ (9)

By solving (7) and (8) we get the eigen vector