I am trying to prove that the following matrix is positive semidefinite: A = [2 1 2; 1 1 1; 2 1 2]. I found that the eigenvalues are 0, 0, and 5. After finding the row echelon form, the pivots are greater than 0. So, this should mean that the matrix is positive semidefinite. But when I try to test the principal submatrices, some of them are less than 0. That means that the matrix is not positive semidefinite.  Am I doing this wrong? Can you prove this is a positive semidefinte matrix with the use of the "No principal submatrices have negative determinants" rule?

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