In this page characteristic roots questions 3 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 3 :
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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= (-2-λ)[ -λ(1-λ) - 12 ] - 2[-2 λ - 6] - 3 [-4-(-1)(1-λ) ]
= (-2-λ)[ -λ + λ² - 12 ] + 4 λ + 12 - 3 [-4+1-λ ]
= (-2-λ)[ λ² -λ - 12 ] + 4 λ + 12 - 3 [-3-λ ]
= (-2-λ) [λ² -λ - 12 ] + 4 λ + 12 + 9 + 3 λ
= -2λ² + 2λ + 24 - λ³ + λ² + 12 λ + 4 λ + 12 + 9 + 3 λ
= - λ³ - λ² + 2λ + 12 λ + 4 λ + 3 λ + 24 + 12 + 9
= - λ³ - λ² + 21λ + 45
= λ³ + λ² - 21λ - 45
To find roots let |A-λI| = 0
λ³ + λ² - 21λ - 45 = 0
For solving this equation first let us do synthetic division.characteristic roots questions 3 characteristic roots questions 3
By using synthetic division we have found one value of λ that is λ = -3.
Now we have to solve λ² - 2 λ - 15 to get another two values. For that let us factorize
λ² - 2 λ - 15 = 0
λ² + 3 λ - 5 λ - 15 = 0
λ (λ + 3) - 5 (λ + 3) = 0
(λ - 5) (λ + 3) = 0
λ - 5 = 0
λ = 5
λ + 3 = 0
λ = - 3
Therefore the characteristic roots (or) Eigen values are x = -3,-3,5
Questions |
Solution |
Question 1 : Determine the characteristic roots of the matrix
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Question 2 : 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 question3 Determine the characteristic roots of the matrix
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characteristic roots questions 3 |
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