The reduced form of the matrix [A | I] is

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

Since the left-hand side is not the identity, it follows that the matrix A is singular (has no inverse).

Note: You might instead have obtained the matrix

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

Although this matrix is not completely reduced (the 3,4 entry does not have its column cleared and is not a 1), you could have stopped here, since the row of three zeros on the left shows that you can never obtain the identity on the left.

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