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    Adjoint of a Matrix

    Result

    (1 | 3 | 2
2 | 2 | 1
3 | 1 | 3)

    Dimensions

    3 (rows) × 3 (columns)

    Matrix plot

    Matrix plot

    Trace

    6

    Determinant

    -12

    Inverse

    1/12(-5 | 7 | 1
3 | 3 | -3
4 | -8 | 4)

    Characteristic polynomial

    -λ^3 + 6 λ^2 + 2 λ - 12

    Eigenvalues

    λ_1 = 6

    λ_2 = -sqrt(2)

    λ_3 = sqrt(2)

    Eigenvectors

    v_1 = (11, 9, 14)

    v_2 = (1/2 (-2 - sqrt(2)), 1/sqrt(2), 1)

    v_3 = (1/2 (-2 + sqrt(2)), -1/sqrt(2), 1)

    Diagonalization

    (1 | 3 | 2
2 | 2 | 1
3 | 1 | 3) = S.J.S^(-1)
where
S = (11 | -1 - 1/sqrt(2) | 1/sqrt(2) - 1
9 | 1/sqrt(2) | -1/sqrt(2)
14 | 1 | 1)
J = (6 | 0 | 0
0 | -sqrt(2) | 0
0 | 0 | sqrt(2))
S^(-1) = (1/34 | 1/34 | 1/34
1/68 (-14 - 9 sqrt(2)) | 1/68 (25 sqrt(2) - 14) | 1/68 (20 - 9 sqrt(2))
1/68 (9 sqrt(2) - 14) | 1/68 (-14 - 25 sqrt(2)) | 1/68 (20 + 9 sqrt(2)))

    Condition number

    7.58333

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