How do you know if a system of N linear equations with N unknowns, has N unique solutions?
Linear equations help
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linear-algebra
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0Ethan, 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
1 Answers
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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:
- An infinite number of solutions.
- No solution.
- 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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0Yes thanks alot – 2012-11-28