Here are the steps in the by-hand row-reduction of the matrix [A | I]:

1     0     0     1     0     0   2 -1 0   0     1     0   R2 - 2R1 1 1 1   0     0     1   R3 - R1

1     0     0     1     0     0   0 -1 0   -2     1     0   0 1 1   -1     0     1   R3 + R2

1     0     0     1     0     0   0 -1 0   -2     1     0   (-1)R2 0 0 1   -3     1     1

1     0     0     1     0     0   0 1 0   2     -1     0   0 0 1   -3     1     1

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