Showing nonsingular matrix has a unique solution

• bonfire09
In summary, the conversation discusses a homework problem that states that a linear system with a nonsingular matrix of coefficients has a unique solution. The problem also mentions representing an arbitrary nxn matrix in echelon form to prove this. The participants also discuss the meaning of "non-singular" and how it relates to the matrix having at least one solution.
bonfire09

Homework Statement

It states that for any linear system with a nonsingular matrix of coefficients has solution and its solution is unique.

The Attempt at a Solution

I wanted to put an diagram of an arbitrary nxn matrix showing it has unique solution but I'm not sure how to represent a nxn matrix in echelon form?

Last edited:
correct...The problem says (i think which you are trying to prove) that it has a solution and the solution is unique. So you know that the matrix of the arbitrary nxn matrix will have a pivot and ever row and column. To represent that use the given that it is a linear system with nonsingular matrix of coefficients. I've never seen the description as "non singular" before but I assume it relates to the fact that the matrix will be isomorphic to r^n

Oh ok. I assume its OK to use explain it using the assumption the matrix is nonsingular(means it has at one solu ion).

What is a nonsingular matrix?

A nonsingular matrix is a square matrix that has a determinant that is not equal to 0. This means that the matrix has an inverse and can be solved for a unique solution.

Why is it important to show that a nonsingular matrix has a unique solution?

It is important to show that a nonsingular matrix has a unique solution because it guarantees that the system of equations represented by the matrix has only one solution. This is useful in many applications, including linear programming, optimization, and engineering problems.

How do you prove that a nonsingular matrix has a unique solution?

To prove that a nonsingular matrix has a unique solution, we can use the determinant of the matrix. If the determinant is not equal to 0, then the matrix is nonsingular and has an inverse. This inverse can then be used to solve for a unique solution.

What happens if a matrix is singular?

If a matrix is singular, it means that the determinant is equal to 0 and the matrix does not have an inverse. This can lead to multiple solutions or no solutions at all for the system of equations represented by the matrix.

Can a nonsingular matrix have more than one solution?

No, a nonsingular matrix can only have one solution. This is because the inverse of a nonsingular matrix is unique, and the inverse is used to find the solution to a system of equations represented by the matrix.

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