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I have an issue with zeroing the 3x3 matrix to find the eigenvector.

I have found the characteristic equation for the 3 eigenvalues.

the matrix is

1 1/2 1/3

1/2 1/3 1/4

1/3 1/4 1/5

The equation i got is -A^3 + (23/15)A^2 - (127/720)A + (1/2160) which correspond to A = 0.1223, 0.0027 & 1.41.

I have tried row swapping, division and multiplication but cannot get the correct eigenvector.

Please help. Thank you.