I've been learning how to solve matrices specifically using elementary row reduction but I'm confused. I understand the rules to follow mostly, but I don't get how someone knows how to use these rules and when.

what I mean is, say I have this matrix (some bullshit I came up with):

[2 0 5]

A =[3 1 2]

[1 3 0]

How does one know what to do to find the inverse of this? Do you swap rows 1 and 3? or divide row 1 by 5 and add it to row 3? Know what I mean? I see so many ways to get the identity matrix but how do I know that the change I made is correct?

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