Linear algebra unique solutions

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A coefficient matrix with a determinant of 0 indicates that it lacks an inverse, resulting in no unique solution for the linear system. A unique solution is defined as one that does not involve free variables, meaning it is a single, specific solution. When the determinant is 0, the system can either have infinitely many solutions or none at all. The discussion emphasizes understanding the implications of the determinant on the nature of solutions in linear algebra. Examples can help clarify these concepts further.
charlies1902
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This is just a general question.

When a coefficient matrix for a linear system has a determinant equal to 0. That means the coefficient matrix does not have an inverse, thus the system does not have a unique solution.

Is the above statement correct?

What exact is a unique solution? Is it basically just one without free variables?
 
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That's essentially correct. Might be best though if you thought about why it's true so you can answer this yourself. If M is your coefficient matrix, then det(M)=0 means Mx=0 has a solution. So Mx=0 has an infinite number of solutions. And the solution not being unique could mean either you have an infinite number of solutions (i.e. free parameters) or you might have no solutions. Can you give an example?
 

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