## Characteristic Roots Questions 5

In this page characteristic roots questions 5 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 5 :

Determine the characteristic roots of the matrix

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

Let A =

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

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

 1 0 0 0 1 0 0 0 1

Now we have to multiply λ with unit matrix I.

λI =

 λ 0 0 0 λ 0 0 0 λ

A-λI=

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

-

 λ 0 0 0 λ 0 0 0 λ

=

 (11-λ) (-4-0) (-7-0) (7-0) (-2-λ) (-5-0) (10-0) (-4-0) (-6-λ)

=

 (11-λ) -4 -7 7 (-2-λ) -5 10 -4 (-6-λ)

A-λI=

 (11-λ) -4 -7 7 (-2-λ) -5 10 -4 (-6-λ)

=  (11-λ)[(-2-λ)(-6-λ)-20]+4[7(-6-λ)+50]-7[-28-10(-2-λ)]

=  (11-λ)[(2+λ)(6+λ)-20]+4[-42-7λ+50]-7[-28+20+10λ]

=  (11-λ)[12+2λ+6λ+λ²-20]+4[8-7λ]-7[-8+10λ]

=  (11-λ)[λ²+8λ-8]+32-28λ+56-70λ

=  11λ²+8λ-88-λ³-8λ²+88λ-98λ+88

=  - λ³+3λ²-2λ

=  -λ³+3λ²-2λ

=  -λ(λ²-3λ²+2)

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

 Questions Solution

Question 1 :

Determine the characteristic roots of the matrix

 5 0 1 0 -2 0 1 0 5

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

Solution

Question 4 :

Determine the characteristic roots of the matrix

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

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