An n x n matrix with two identical rows has infinitely many solutions

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Homework Help Overview

The discussion revolves around proving that an n x n matrix with two identical rows has infinitely many solutions, specifically focusing on the implications of row reduction without invoking concepts like rank or nullity.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants explore the consequences of subtracting one identical row from another, leading to a zero row. Questions arise about the implications of this zero row on the number of equations and variables.

Discussion Status

The discussion is active, with participants examining the relationship between the number of equations and variables after row reduction. Some guidance has been offered regarding the implications of having a zero row, but no consensus has been reached on the overall proof.

Contextual Notes

There is an emphasis on not using advanced concepts such as rank or nullity, which may limit the approaches discussed. The urgency expressed by the original poster suggests a time constraint for the homework task.

Jamin2112
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Homework Statement



Prove the thread title.

Homework Equations



Without using anything about rank, nullity, etc., --- just row reduction

The Attempt at a Solution



Can't figure this out. It's actually a friend one of my buddies is doing for his class. PLEASE ANSWER IN LESS THAN 2 HOURS! IT'S URGENT!
 
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Well what will it mean if you subtract identical row 1 from the other identical row?
 
rock.freak667 said:
Well what will it mean if you subtract identical row 1 from the other identical row?

One of the rows will now be zero
 
infinitely many solutions to what?
 
lanedance said:
infinitely many solutions to what?

Ax = 0 where A is an n x n matrix
 
OK, so one of the rows is now zero, meaning the only information it gives us is 0=0. How many equations do you now have, and in how many variables?
 
n - 1 equations and n variables ... meaning an infinite number of solutions.
 

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