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16 July, 03:16

Find the eigenvectors and eigenvalues for the matrix. A =

[2 - 2

-4 4]

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Answers (1)
  1. 16 July, 04:02
    0
    Consider lambda as $

    $ = 0, & $ = 6 (eigen values)

    eigen vectors:

    [1 2]

    [0 0]

    Step-by-step explanation:

    1st step:

    Lambda I

    here I is identity matrix.

    As our matrix is 2X2, so we will take I = [ 1 0, 0 1]

    consider lambda is represented as $

    so $I matrix will be [$ 0, 0 $]

    2nd Step:

    A - Lambda I

    i. e.[2 - 2, - 4 4] - [$ 0, 0 $]

    => [2-$ - 2, - 4 4-$]

    3rd Step:

    Det [A-$I]

    i. e. [ (2-$) (4-$) - (-2) (-4) ] = 0

    =>8-2$-4$+$²-8

    =>$²-6$

    4th Step:

    Det [A-$I]=0

    i. e. $²-6$ = 0

    $ ($-6) = 0

    i. e. $ = 0, & $ = 6 (eigen values)

    5th Step

    put these eigen values in [A-$I] matrix

    i. e if we put $ = 0

    we get [2 - 2, - 4 4]

    consider this matrix as B

    then

    B X⁻ = 0⁻ (⁻ is a bar sign notation)

    [2 - 2, - 4 4] [x₁ x₂] = [0 0]

    by row reduction

    -2R₁ + R₂ - > R₂

    so

    2X₁-2X₂=0

    X₁=X₂

    ie eigen vector will be [0 0]

    now consider

    i. e if we put $ = 6

    we get [-4 - 2, - 4 - 2]

    consider this matrix as B

    then

    B X⁻ = 0⁻ (⁻ is a bar sign notation)

    [-4 - 2, - 4 - 2] [x₁ x₂] = [0 0]

    by row reduction

    R₁ - R₂ - > R₂

    so

    -4X₁-2X₂=0

    2X₁=X₂

    if we put X₁=1

    x₂=2

    So eigen vector will be

    [1 2]
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