Calculation of eigenvalues and eigenvectors

Questions
1

I want to calculate the eigenvalues and eigenvectors of matrix A. I did it with two methods.

Method A: I solved the expressions of eigenvalues and eigenvectors of Matrix A firstly, and then specific values were substituted for the solutions.

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Method B: Specific values were substituted into Matrix A firstly, and then its eigenvalues and eigenvectors were solved.

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However, I got different results with these two methods. I wonder what the reason is for the difference and which result is right.

Mathod A.nb (44.9 KB)
Mathod B.nb (8.92 KB)

2

It seems method A divided to get third component of 1 in the eigenvectors. For this particular substitution that does not work because a denominator vanishes in the second eigenvectors. Hence the large values in the other two components, from division by an approximate zero.