Characteristic Roots Question 1

In this page characteristic roots question1 we are going to see how to find characteristic roots of any given matrix.

Definition :

Let A be any square matrix of order n x n and I be a unit matrix of same order. Then |A-λI| is called characteristic polynomial of matrix. 

Then the equation |A-λI| = 0 is called characteristic roots of matrix.  The roots of this equation is called characteristic roots of matrix.

Another name of characteristic roots:

characteristic roots are also known as latent roots or eigenvalues of a matrix.

Question 1 :

Determine the characteristic roots of the matrix

5 0 1 0 -2 0 1 0 5

Solution:

Now we have to multiply λ with unit matrix I.

To find roots let |A-λI| = 0

   λ³ - 8 λ² + 4 λ + 48 = 0

For solving this equation first let us do synthetic division.characteristic roots question1

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 - 10 λ + 24 = 0

λ² - 6 λ - 4 λ + 24 = 0

λ (λ - 6) - 4 (λ - 6) = 0

(λ - 6) (λ - 4) = 0

 λ - 6 = 0

 λ = 6

 λ - 4 = 0

 λ = 4

Therefore the characteristic roots (or) Eigen values are x = -2,4,6

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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
  

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