When does a system of equations have infinite, unique and no solutions

Question

Can someone please tell me what a matrix looks like when there is infinite solutions, unique solution and no solutions. I have been searching the internet and I cannot find a straightforward answer of what the matrix should look like. The question I am faced with is that I have a variable in a 3x3 matrix and I have to get the values of that variable when there is no solution, infinite and unique solution.

Answer 1

Let us start with the simpler case of a system of $2$ equations in $2$ unknowns.

$$\begin{cases}x+y=3\\x-y=1\end{cases}$$ has a unique solution because it is equivalent to

$$\begin{cases}x+y=3\\2x=4\end{cases}$$ by adding the first equation to the second. Then $x$ is uniquely determined and so is $y$.

Now,

$$\begin{cases}x+y=3\\x+y=1\end{cases}$$ is equivalent to

$$\begin{cases}x+y=3\\0=-2\end{cases}$$ by subtracting the first from the second, and this system has no solution, as the second equation is impossible.

And finally

$$\begin{cases}x+y=3\\x+y=3\end{cases}$$ is equivalent to

$$\begin{cases}x+y=3\\0=0\end{cases}$$ which has an infinity of solutions as the second equation is always true.

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The approach generalizes to larger systems. If, by clever combinations of the equations, you obtain always-false or always-true equations, then the system is impossible or indeterminate, respectively.

There is a systematic method to combine the equations in a way that progressively forms smaller systems, called Gaussian elimination. It will transform a square system in a triangular one. If at some stage all remaining coefficients are zero, then you are in one of these singular cases.