Linear algebra unique solutions

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SUMMARY

A coefficient matrix in a linear system with a determinant equal to 0 indicates that the matrix lacks an inverse, confirming that the system does not possess a unique solution. A unique solution is defined as one that does not involve free variables, implying that the system has a single solution. When the determinant is zero, the system either has infinitely many solutions or no solutions at all, depending on the specific conditions of the equations involved.

PREREQUISITES
  • Understanding of linear systems and matrices
  • Knowledge of determinants and their implications
  • Familiarity with the concept of unique solutions in linear algebra
  • Basic grasp of free variables in the context of linear equations
NEXT STEPS
  • Study the properties of determinants in linear algebra
  • Learn about the implications of matrix inverses on linear systems
  • Explore examples of linear systems with unique, infinite, and no solutions
  • Investigate the role of free variables in determining the nature of solutions
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to clarify concepts related to linear systems and their solutions.

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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