Proof of Nonsingular Matrices: Linear Algebra

In summary, the conversation is about explaining whether a given proof is sufficient. The person's thought process was to start with the definition of a nonsingular matrix and use it to prove that AB is also nonsingular and (AB)- = A- - B-. They found a different proof online and are asking if their approach is incorrect. The expert summarizer points out that non-singular matrices have an inverse A-1 such that AA-1 = A-1A = I and (AB)-1 = B-1A-1, not A-1B-1.
  • #1
Mdhiggenz
327
1

Homework Statement


http://i48.tinypic.com/2qu14ax.jpg

Can you guys explain whether or not my proof would be sufficient. My thought process was to start with the definition.

So if A is nonsingular is means that A has an inverse such that

A=A-. I used that same thinking for B. B=B-

Then using the question which states. IF A and B are nonsingular nxn matrices then AB is also nonsingular and (AB)-=A-B-

I thought it would be easier to prove the right side so I started with AB=A-B-

and used the relationship A=A- and B=B- to show that both sides are equal. However I found the proof online, and they did something slightly different.

Am I incorrect, if so where did I go wrong in my thinking?

Thanks

Homework Equations


The Attempt at a Solution

 
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  • #2
Non-singular does not mean that A = A-1

It means that there is an inverse A-1 such that AA-1 = A-1A = I
 
  • #3
And (AB)-1 = B-1A-1, not A-1B-1
 

FAQ: Proof of Nonsingular Matrices: Linear Algebra

1. What is a nonsingular matrix?

A nonsingular matrix is a square matrix that has a nonzero determinant, meaning it has a unique inverse and is therefore invertible. This is also known as a regular or non-degenerate matrix.

2. How do you prove a matrix is nonsingular?

To prove a matrix is nonsingular, you can use several methods such as the determinant method, where you calculate the determinant of the matrix and if it is nonzero, the matrix is nonsingular. You can also use the inverse method, where you find the inverse of the matrix and if it exists, the matrix is nonsingular. Other methods include the rank method and the eigenvalue method.

3. What does it mean for a matrix to be singular?

A singular matrix is a square matrix that has a determinant of zero, meaning it does not have a unique inverse and is therefore not invertible. This is also known as a degenerate matrix.

4. Can a nonsingular matrix become singular?

No, a nonsingular matrix cannot become singular. The only way a matrix can become singular is if it was already singular to begin with or if it is transformed into another singular matrix through operations such as row reduction.

5. Why is it important to determine if a matrix is nonsingular?

Determining if a matrix is nonsingular is important because nonsingular matrices have many useful properties and can be used to solve linear equations and systems of equations. They also have applications in areas such as computer graphics, engineering, and economics.

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