Non-Singular matrix - RREF proof

• Soupy11
In summary: Then, consider the implications of a non-singular matrix being in reduced row echelon form. In summary, to prove that the only non-singular reduced row echelon matrix is I sub n, you must start by defining row echelon and reduced row echelon matrices, and then consider the implications of a non-singular matrix being in reduced row echelon form.
Soupy11

Homework Statement

Prove that the only n x n non-singular reduced row echelon matrix is I sub n.

The Attempt at a Solution

Not even remotely sure where to start here - the statement looks similar to the definition of a non-singular matrix. Yet there is something subtly different and I am having an issue grasping it. I see in the next chapter there are some tools explored using elementary matrices, but this specific question is before that material so I am assuming that the proof must be done without that knowledge.

If A is a nxn matrix not equal to I yet non-singular

There exists a matrix X that satisfies
AX=I

Soupy11 said:

Homework Statement

Prove that the only n x n non-singular reduced row echelon matrix is I sub n.

The Attempt at a Solution

Not even remotely sure where to start here - the statement looks similar to the definition of a non-singular matrix. Yet there is something subtly different and I am having an issue grasping it. I see in the next chapter there are some tools explored using elementary matrices, but this specific question is before that material so I am assuming that the proof must be done without that knowledge.

If A is a nxn matrix not equal to I yet non-singular

There exists a matrix X that satisfies
AX=I

Well, if it is non-singular doesn't that mean you won't loose a pivot when you simplify? Thus what would this mean if we simplified all the way?

You probably want to start with the definition of row echelon matrix and reduced row echelon matrix.

1. What is a non-singular matrix?

A non-singular matrix is a square matrix where the determinant is non-zero. This means that the matrix is invertible and has a unique solution to the system of equations it represents.

2. What does RREF stand for?

RREF stands for Reduced Row Echelon Form. It is a way of organizing a matrix to easily solve a system of linear equations by using elementary row operations.

3. How do you prove that a matrix is non-singular using RREF?

The proof involves showing that the matrix, when transformed into RREF, has a pivot in every row. This means that the matrix has linearly independent columns, and therefore has a non-zero determinant.

4. Why is it important to prove that a matrix is non-singular?

A non-singular matrix has a unique solution to its system of equations. This is important in many applications, such as solving equations in physics, engineering, and economics. It also allows for efficient computation and avoids errors in calculations.

5. Can a non-singular matrix become singular?

No, a non-singular matrix cannot become singular. The determinant of a non-singular matrix is always non-zero, and any elementary row operations performed on the matrix will not change the determinant. Therefore, a non-singular matrix will always remain non-singular.

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