Linear algebra unique solutions

In summary, A coefficient matrix with a determinant equal to 0 does not have an inverse and therefore the linear system does not have a unique solution. This means that either there are an infinite number of solutions or there are no solutions at all.
  • #1
charlies1902
162
0
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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  • #2
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?
 

Related to Linear algebra unique solutions

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces and matrices. It involves the study of linear transformations and their properties.

2. What does it mean for a linear system to have a unique solution?

A linear system has a unique solution when there is only one set of values for the variables that satisfies all of the equations in the system. In other words, there is only one point where all of the equations intersect.

3. How do you determine if a linear system has a unique solution?

A linear system has a unique solution if the number of equations is equal to the number of variables and the system is consistent (meaning there is a solution). This can be determined by using methods such as Gaussian elimination or matrix operations to reduce the system to row-echelon form.

4. Can a linear system have more than one solution?

Yes, a linear system can have infinitely many solutions or no solutions at all. This depends on the number of equations and variables and the consistency of the system. If the number of equations is less than the number of variables, there can be infinitely many solutions. If the system is inconsistent, there will be no solutions.

5. How is linear algebra used in real-world applications?

Linear algebra has many practical applications, such as in computer graphics, data analysis, economics, and engineering. It is used to solve systems of equations, perform transformations, and make predictions based on data. It also has applications in fields such as physics, chemistry, and biology.

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