Properties of Singular Matrices

In summary, the conversation discusses whether the product of two singular matrices, A and B, is also singular. It is stated that according to the Lemma, if either A or B is singular, then AB is also singular. However, it is not explicitly stated what happens when both A and B are singular. The person has tried using various examples of singular matrices and found that the product is always singular. They also mention using the determinant to prove this, stating that since both A and B have a determinant of 0, the determinant of AB is also 0, making AB singular. Therefore, it can be concluded that if at least one of the matrices, A or B, is singular, then their product, AB, will also be
  • #1
simmonj7
66
0
1. Homework Statement
State whether true or false:
If A and B are singular matrices, then AB is also singular.



3. The Attempt at a Solution
I know that according to the Lemma, if A or B is a singular matrix, then its product AB is also singular. However, it doesn't speak to what happens if two both A and B are singular. I have tried examples using a bunch of singular matrices which I made, and all turned out to be singular. However, I can't get rid of this gut feeling that maybe there is an exception to this situation which I just can't put my finger.
 
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  • #2
In general, if at least one of them is singular then AB is singular, and that's good enough to prove it.
Another way to prove this is using determinant.
Since |A|=0 and |B|=0 (A and B are singular), we get |AB|=|A||B|=0 => AB is singular too.
 

1. What are singular matrices?

Singular matrices are square matrices that do not have an inverse. This means that they cannot be multiplied by another matrix to produce the identity matrix. In other words, they do not have a unique solution when used to solve a system of linear equations.

2. How do you identify if a matrix is singular?

A matrix is singular if its determinant is equal to zero. The determinant is a mathematical operation that can be performed on a square matrix. If the determinant is equal to zero, then the matrix is singular. Otherwise, it is non-singular.

3. What are the properties of singular matrices?

Some properties of singular matrices include:

  • They do not have an inverse
  • Their determinant is equal to zero
  • They have at least one zero eigenvalue
  • They do not have a unique solution when used to solve a system of linear equations
  • They have linearly dependent rows or columns

4. How are singular matrices used in real-world applications?

Singular matrices are used in various fields such as physics, engineering, and economics. They are used to model systems that have multiple variables and constraints, and their properties help in analyzing the behavior and stability of these systems. In physics, they are used to study the equilibrium of forces in a system, while in economics, they are used to analyze supply and demand equations.

5. Can a singular matrix be converted into a non-singular matrix?

No, a singular matrix cannot be converted into a non-singular matrix. The determinant of a matrix is a fundamental property and cannot be changed by any matrix operation. However, it is possible to transform a singular matrix into a non-singular matrix by adding or subtracting a small value to the diagonal entries. This process is known as matrix regularization.

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