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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

    Transpose

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

    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) = P.D.P^(-1)
where
P≈(11 | -1.70711 | -0.292893
9 | 0.707107 | -0.707107
14 | 1 | 1)
D≈(6 | 0 | 0
0 | -1.41421 | 0
0 | 0 | 1.41421)
P^(-1)≈(0.0294118 | 0.0294118 | 0.0294118
-0.393058 | 0.314049 | 0.106942
-0.018707 | -0.725814 | 0.481293)

    Condition number

    7.58333

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