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How do you know if a system of N linear equations with N unknowns, has N unique solutions?

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    Ethan, For example, the system of equations $\{x+y=0,x-y=0\}$ has the unique solution $(x,y)=(0,0)$ (if the characteristic is not $2$).2012-11-28

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Okay, think of a system of two linear equations in two unknowns, $x$, $y$. If a unique solution exists, then there exist a unique $m, n$ such that $x = m, \; y = n$. That is the unique solution. This solution might be expressed as the ordered pair $(m, n)$.

For any system of $n$ linear equations in $n$ unknowns, one and only one of the following is true:

The system of linear equations has:

  1. An infinite number of solutions.
  2. No solution.
  3. A unique solution.

If more than one solution exists, then an infinite number of solutions exist.

If no solution exists, then the linear system is "inconsistent."

A unique solution exists if and only if the determinant of the coefficient matrix is nonzero.

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    Yes thanks alot2012-11-28