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If I want to row reduce a matrix over lets say $\Bbb{Z}_7$.

Can I row reduce the matrix to echelon form over $\Bbb{R}$ and then manually row reduce the entries using modulo 7?

Thank you.

Rodrigo de Azevedo
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Dvir Peretz
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    Well, as long as the entries of the matrix make sense modulo $;7;$ , yes: you can do that...and as long as you don't use an elementary operation which is forbidden in $;\Bbb F_7;$ , like say multiplying a row by $;\cfrac17;$ ... – DonAntonio Oct 24 '19 at 20:25
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    It’s really much easier to do it in $\Bbb Z_7$. All you need is comfort in multiplying and dividing elements of the field. – Lubin Oct 24 '19 at 22:05
  • What Lubin said. Also, doing all the steps with modular arithmetic makes it easier for you in the end. Otherwise you may end up doing row operations with more complicated numbers. Like you need to multiply a row with $103/29$, and then subtract it from another multiplied by $336/85$. For paper and pencil work I would do operations involving $5$ and $0$ respectively any day of the week. – Jyrki Lahtonen Oct 25 '19 at 05:24
  • Also, you should never leave $\Bbb{Q}$, for then going back to $\Bbb{Z}_7$ will be more taxing to justify. – Jyrki Lahtonen Oct 25 '19 at 05:25
  • @Dvir Here I gave an example in the field $\Bbb{Z}_{29}$. Modulo seven it works the same way, only it is easier because the numbers stay smaller. – Jyrki Lahtonen Oct 27 '19 at 06:03

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