# What fraction of square matrices are singular?

• Bavid
In summary, there are countably infinitely many 3x3 matrices with integer entries, some of which are singular. However, there is no way to get a fraction or percentage of which are singular. The probability of a randomly chosen 3x3 matrix being singular is zero.
Bavid
I was wondering what fraction of 3*3 square matrices are singular? I guess if the elements are real then the answer will be vanishingly small.
If however, the elements are integers, is there a way to work this number out?

I don't think there is any way to get a fraction or percentage. There are countably infinitely many 3x3 matrices with integer entries. And countably many of them are singular. Your "fraction" would be ##\frac \infty \infty##.

LCKurtz said:
I don't think there is any way to get a fraction or percentage. There are countably infinitely many 3x3 matrices with integer entries. And countably many of them are singular. Your "fraction" would be ##\frac \infty \infty##.

I don't know the answer to the original question, but this statement seems a poor argument that such a fraction cannot be well defined. There is a well defined sense in which half of all integers are even, despite the sets of both integers and even integers being countably infinite.

Admittedly the limits might be a little trickier to handle in this case.

@kurros: Yeah, if the question is posed as
'What is the probablility that a given 3*3 matrix with integer entries is singular?" then the answer is either between 0 and 1 or tending to zero.
It is zero if for every singular matrix, one can find an infinite number of unique non-singular matrices. But I cannot think of a way to prove that.

3x3 matrices over a field form a 9 dimensional vector space over that field and singular ones form an 8 dimensional non linear subvariety of degree 3, defined by setting the determinant equal to zero. a lower dimensional subvariety has density zero if the field of definition is algebraically closed. But if you work over a finite field then you can compute the cardinality of both sets and find a fraction. But over the complex numbers, the probability of a randomly chosen 3x3 matrix being singular is zero.

Bavid said:
I was wondering what fraction of 3*3 square matrices are singular? I guess if the elements are real then the answer will be vanishingly small.
If however, the elements are integers, is there a way to work this number out?

Hey Bavid.

One suggestion I have for you if you want to pursue this is to first restrict the range to a finite number of cases.

For simplicity let N be a fixed number. Then consider all the 3x3 matrices with 0 < X < N where any element may take on the value of X in your matrix.

From this you can start to get empirical data which might give a pattern for some function of N. From this you could proceed to find a relationship of some sort for N.

I would also suggest that you make use of number theory since for a singular matrix, at least one row has to be a linear combination of the other and I imagine some parts of number theory could help you analyze these kinds of situations since we are talking about linear relationships over say any non-linear relationship.

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

A singular matrix is a square matrix that does not have an inverse. This means that the matrix cannot be multiplied by another matrix to get the identity matrix. In other words, the determinant of a singular matrix is equal to 0.

## 2. How is singularity determined in a square matrix?

Singularity is determined by calculating the determinant of the square matrix. If the determinant is equal to 0, then the matrix is singular. Otherwise, it is non-singular.

## 3. What is the difference between a singular and non-singular matrix?

A non-singular matrix has an inverse, meaning it can be multiplied by another matrix to get the identity matrix. A singular matrix does not have an inverse and cannot be multiplied by another matrix to get the identity matrix.

## 4. Are there any real-world applications for singular matrices?

Yes, singular matrices are commonly used in linear algebra and have various applications in fields such as physics, engineering, and economics. For example, they can be used to solve systems of linear equations and to analyze the stability of dynamic systems.

## 5. What fraction of square matrices are singular?

The fraction of singular matrices among all possible square matrices depends on the size of the matrix. For a 2x2 matrix, approximately 1/3 of all matrices are singular. For a 3x3 matrix, approximately 1/4 are singular. As the size of the matrix increases, the fraction of singular matrices approaches 0.

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