Assume that . Let us show that is row equivalent to . Assume moreover that . Then divide the first row by to get the new matrix
Now take the second row minus times the first row to get
Divide the second row by
since it not equal to 0 to get
Finally take the first row minus
times the first row to get
Our proof is almost complete, if we show that the conclusion still holds when . In this case, neither nor are equal to 0. Switch the first row with the second one to get
Divide the first row by and the second row by to get
Take the first row minus
times the second row to get
The proof is now complete.
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