Is Matrix A Invertible? Solving Linear System with Gaussian Elimination

However, since there are infinitely many solutions, it follows that A cannot be invertible.In summary, the fact that the linear system Ax=c has infinitely many solutions, while an invertible matrix should have exactly one solution, suggests that the matrix A is not invertible.
  • #1
gpax42
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Homework Statement



Suppose that Gaussian Elimination gives the soluiton of a Linear System Ax=c as x = x0 + a1x1 + a2x2, where A is a 6X6 matrix and a1 and a2 are arbitrary. Is the matrix A invertible? Explain


The Attempt at a Solution



I simply explained that due to properties of an invertible matrix, Ax=c must have exactly one solution but if a1 and 2 are arbitrary, then x = x0 + a1x1 + a2x2 has infinitely many solutions and A cannot be invertible

does anyone agree with this answer and if not, any advice on how to start the problem would be great appreciated :redface: thanks a lot!
 
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  • #2
I'm assuming that x0, x1, and x2 are vectors. Your reasoning looks fine to me. If A had been invertible then you would have gotten a unique solution, which could be written as x = A-1c.
 

1. What is a linear system problem?

A linear system problem is a mathematical problem that involves solving a set of equations with multiple variables. These equations are linear, meaning that the variables are raised to the first power and are not multiplied together. The goal of solving a linear system problem is to find the values of the variables that satisfy all of the equations simultaneously.

2. What are the different methods for solving a linear system problem?

There are several methods for solving a linear system problem, including substitution, elimination, and graphing. Substitution involves solving for one variable in terms of the others and then plugging this value into the other equations. Elimination involves adding or subtracting equations to eliminate one variable at a time. Graphing involves plotting the equations on a graph and finding the point of intersection, which represents the solution to the system.

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

A linear system problem has a unique solution if the number of equations is equal to the number of variables, and the equations are independent, meaning they are not multiples of each other. This means that each equation contains unique information and can help to determine the values of the variables. If the number of equations is less than the number of variables, the system may have infinite solutions or no solutions at all.

4. Can a linear system problem have no solution?

Yes, a linear system problem can have no solution if the equations are inconsistent, meaning they contradict each other and cannot be satisfied simultaneously. This can happen if the equations represent parallel lines or if there are more variables than equations. Inconsistent systems have no solution because there is no set of values that can satisfy all of the equations at the same time.

5. How are linear system problems used in real life?

Linear system problems have many practical applications in fields such as engineering, economics, and physics. They can be used to model and solve real-world problems involving multiple variables, such as determining the optimal production levels for a company or calculating the trajectory of a projectile. Linear systems are also used in data analysis and machine learning to find patterns and make predictions based on multiple variables.

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