## 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:

Let A =

 5 0 1 0 -2 0 1 0 5

The order of A is 3 x 3. So the unit matrix I =

 5 0 1 0 -2 0 1 0 5

Now we have to multiply λ with unit matrix I.

λI =

 λ 0 0 0 λ 0 0 0 λ

A-λI=

 5 0 1 0 -2 0 1 0 5

-

 λ 0 0 0 λ 0 0 0 λ

=

 (5-λ) (0-0) (1-0) (0-0) (-2-λ) (0-0) (1-0) (0-0) (5-λ)

=

 (5-λ) 0 1 0 (-2-λ) 0 1 0 (5-λ)

A-λI=

 (5-λ) 0 1 0 (-2-λ) 0 1 0 (5-λ)

=  (5-λ)[(-2-λ) (5-λ) - 0] - 0 [(0 - 0)] + 1 [0- (-2 -λ)]

=  (5-λ)[ -10 + 2 λ - 5 λ + λ²] - 0 + 2 + λ

=  (5-λ)[ -10 - 3 λ + λ²] - 0 + 2 + λ

=  (5-λ)[λ² -3 λ-10] + 2 + λ

=  5 λ² - 15 λ - 50 - λ³ + 3 λ² + 10 λ + 2 + λ

= - λ³ + 5 λ² + 3 λ² - 15 λ + 10 λ + λ - 50 + 2

= - λ³ + 8 λ² - 4 λ - 48

=  λ³ - 8 λ² + 4 λ + 48

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

 Questions Solution

Question 2 :

Determine the characteristic roots of the matrix

 1 1 3 1 5 1 3 1 1

Question 3 :

Determine the characteristic roots of the matrix

 -2 2 -3 2 1 -6 -1 -2 0

Question 4 :

Determine the characteristic roots of the matrix

 4 -20 -10 -2 10 4 6 -30 -13

Question 5 :

Determine the characteristic roots of the matrix  characteristic roots question 1  characteristic roots question 1

 11 -4 -7 7 -2 -5 10 -4 -6

characteristic roots question 1 characteristic roots question 1

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