## Characteristic Roots Questions 4

In this page characteristic roots questions 4 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 4 :

Determine the characteristic roots of the matrix

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

Let A =

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

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=

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

-

 λ 0 0 0 λ 0 0 0 λ

=

 (-4-λ) (-20-0) (-10-0) (-2-0) (10-λ) (4-0) (6-0) (-30-0) (-13-λ)

=

 (-4-λ) -20 -10 -2 (10-λ) 4 6 -30 (-13-λ)

= (4-λ)[(10-λ)(-13- λ)+120]+

20[-2(-13-λ)-24]-10[60-6(10-λ)]

= (4-λ)[-130-10 λ+13λ+λ²+120]+20[26+2λ-24]-10[60-60+6λ]

= (4-λ)[-10+3λ+λ²]+20[2+2λ]-10[6λ]

= (4-λ)[λ²+3λ-10]+20[2+2λ]-10[6λ]

= 4λ²+12λ-40-λ³-3λ²+10λ+40λ+40-60λ

= -λ³ + 1λ² + 2λ

To find roots let |A-λI| = 0

-λ³ + 1λ² + 2λ = 0

For solving this equation -λ from all the terms characteristic roots question4

-λ (λ² - 1λ - 2) = 0

-λ = 0 (or) λ² - 1 λ - 2 = 0

λ = 0       (λ+1) (λ-2) = 0

λ + 1 = 0       λ - 2 = 0

λ = - 1            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

characteristic roots questions 4  characteristic roots questions 4 characteristic roots questions 4

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 5 :

Determine the characteristic roots of the matrix

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

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It divides sorrow and multiplies forgiveness and love.

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