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**bubble86** Let A be an nxn matrix (not assumed to be invertible). Show that there exists an invertible nxn matrix P such that

APA = A

if such matrix existed then wouldn't it mean PA = I ( nxn identity matrix) and

AP = I hence A is invertible.

but anyways here are my ideas so far on this.

I am guessing P would be an elementary matrix (i.e. the ones concerned with row / column operations) if you left multiple by these elementary matrices you are doing row operations and if you right multiply you are doing column operations.

any sort of helpful hints would be much appreciated.