Can't find my mistake trying to find the inverse:
When you do $R_2 \to R_2-2R_3$ on the 2nd page, you get in the 1st column $\frac23-2\times\frac19=\frac13$, while it should be $\frac49$.
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When you do $R_2 \to R_2-2R_3$ on the 2nd page, you get in the 1st column $\frac23-2\times\frac19=\frac13$, while it should be $\frac49$.