In this page characteristic roots question2 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 2 :
Determine the characteristic roots of the matrix
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Let A = |
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The order of A is 3 x 3. So the unit matrix I = |
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Now we have to multiply λ with unit matrix I.
λI = |
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A-λI= |
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- |
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  = |
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  = |
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A-λI= |
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= (1-λ)[ (5-λ)(1-λ) - 1 ] - 1[1 - λ - 3] + 3 [1 - 3 (5-λ) ] = (1-λ)[ 5 - 5 λ - λ + λ² - 1 ] - 1[ -λ - 2] + 3 [ 1 - 15 +3 λ ] = (1-λ)[ λ² - 6 λ + 4 ] + 1 λ + 2 + 3 [ - 14 +3 λ ] = λ² - 6 λ + 4 - λ³ + 6 λ² - 4 λ + λ + 2 - 42 + 9 λ = - λ³ + λ² + 6 λ² - 6 λ - 4 λ + λ + 9 λ + 4 + 2 - 42 = - λ³ + 7 λ² - 10 λ + 10 λ + 6 - 42 = - λ³ + 7 λ² - 36 = λ³ - 7 λ² + 36 |
To find roots let |A-λI| = 0
λ³ - 7 λ² + 36 = 0
For solving this equation first let us do synthetic division.characteristic roots question 2 characteristic roots question 2 characteristic roots question 2 characteristic roots question 2
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 - 9 λ + 18 = 0
λ² - 3 λ - 6 λ + 18 = 0
λ (λ - 3) - 6 (λ - 3) = 0
(λ - 6) (λ - 3) = 0
λ - 6 = 0
λ = 6
λ - 3 = 0
λ = 3
Therefore the characteristic roots (or) Eigen values are x = -2,3,6
Questions |
Solution |
Question 1 : Determine the characteristic roots of the matrix
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Question 3 : characteristic roots question2 Determine the characteristic roots of the matrix
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Question 4 : Determine the characteristic roots of the matrix
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Question 5 : characteristic roots question2 Determine the characteristic roots of the matrix
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Sep 29, 23 10:55 PM
Sep 29, 23 10:49 PM
Sep 29, 23 07:56 PM