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
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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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= (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
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characteristic roots questions 4 characteristic roots questions 4 characteristic roots questions 4 | |||||||||||||
Question 2 : Determine the characteristic roots of the matrix
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Question 3 : Determine the characteristic roots of the matrix
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Question 5 : Determine the characteristic roots of the matrix
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characteristic roots question4 |
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