In this page characteristic vectors of matrix 3 we are going to see how to find characteristic equation of any matrix with detailed example.
Definition :
The eigen vector can be obtained from (A- λI)X = 0. Here A is the given matrix λ is a scalar,I is the unit matrix and X is the columns matrix formed by the variables a,b and c.
Another name of characteristic Vector:
Characteristic vector are also known as latent vectors or Eigen vectors of a matrix.
Question 3 :
Determine the characteristic vector of the matrix
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To find eigen vector first let us find characteristic roots of the given matrix.
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.
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
Substitute λ = -3 in the matrix A - λI
                = |
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From this matrix we are going to form three linear equations using variables x,y and z.
1x + 2y - 3z = 0 ------ (1)
2x + 4y - 6z = 0 ------ (2)
-1x - 2y + 3z = 0 ------ (3)
By solving (1) and (3) we get the eigen vector
The eigen vector x = |
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Substitute λ = 5 in the matrix A - λI characteristic vectors of matrix 3
  = |
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From this matrix we are going to form three linear equations using variables x,y and z.
-7x + 2y - 3z = 0 ------ (4)
2x - 4y - 6z = 0 ------ (5)
-1x - 2y - 5z = 0 ------ (6)
By solving (4) and (5) we get the eigen vector characteristic vectors of matrix3
The eigen vector z = |
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Questions |
Solution |
Question 1 : characteristic vectors of matrix3 Determine the characteristic vector of the matrix
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Question 2 : Determine the characteristic vector of the matrix
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Question 4 : Determine the characteristic vector of the matrix
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characteristic vectors of matrix 3 characteristic vectors of matrix 3 characteristic vectors of matrix 3 characteristic vectors of matrix 3 | |||||||||||||
Question 5 : Determine the characteristic vector of the matrix
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Apr 01, 23 11:43 AM
Mar 31, 23 10:41 AM
Mar 31, 23 10:18 AM