Factorising The Determinant

Introduction

In the realm of linear algebra, matrices and determinants play a crucial role in solving systems of equations and understanding the properties of linear transformations. The determinant of a matrix is a scalar value that can be used to determine the invertibility of the matrix, as well as the volume scaling factor of the linear transformation represented by the matrix. In this article, we will delve into the concept of factorising the determinant, exploring the possibility of expressing a given matrix as the product of two other matrices with integer coefficients.

The Problem Statement

Let $A$ be any matrix with integer coefficients such that $\det(A) = n$. The question arises: can we find two matrices with integer coefficients, namely $U$ and $V$, such that:

$$UV = A$$

$$\det(U) = p$$

$$\det(V) = q$$

where $p \neq 1 \neq q$?

Understanding the Determinant

Before we proceed, let's take a moment to understand the concept of the determinant. The determinant of a matrix is a scalar value that can be calculated using various methods, including the expansion by minors, the cofactor expansion, or the use of a determinant formula. For a matrix $A$ with integer coefficients, the determinant is also an integer.

The Factorisation Problem

The problem of factorising the determinant can be seen as a search for two matrices, $U$ and $V$, such that their product is equal to the original matrix $A$. This is a classic problem in linear algebra, and it has been extensively studied in the context of matrix factorisation.

A Possible Approach

One possible approach to solving this problem is to use the concept of matrix similarity. Two matrices, $A$ and $B$, are said to be similar if there exists an invertible matrix $P$ such that $A = PBP^{-1}$. This concept is useful in understanding the properties of matrices and their determinants.

The Role of the Determinant

The determinant plays a crucial role in the factorisation problem. If we can find two matrices, $U$ and $V$, such that their product is equal to the original matrix $A$, then the determinant of $A$ must be equal to the product of the determinants of $U$ and $V$. This is a fundamental property of determinants, and it can be used to derive constraints on the possible values of the determinants of $U$ and $V$.

Constraints on the Determinants

Let's assume that we have found two matrices, $U$ and $V$, such that their product is equal to the original matrix $A$. Then, we can write:

$$\det(A) = \det(UV) = \det(U)\det(V)$$

Since the determinant of $A$ is equal to $n$, we can write:

$$n = \det(U)\det(V)$$

This equation imposes a constraint on the possible values of the determinants of $U$ and $V$. Specifically, the product of the determinants of $U$ and $V$ must be equal to $n$.

The Case of $n = 1$

If $n = 1$, then the determinant of $A$ is equal to 1. In this case, we can write:

$$1 = \det(U)\det(V)$$

This equation implies that either $\det(U) = 1$ or $\det(V) = 1$. However, this is not possible, since we are given that $p \neq 1 \neq q$. Therefore, the case of $n = 1$ is not possible.

The Case of $n > 1$

If $n > 1$, then the determinant of $A$ is greater than 1. In this case, we can write:

$$n = \det(U)\det(V)$$

This equation implies that either $\det(U) > 1$ or $\det(V) > 1$. However, this is not possible, since we are given that $p \neq 1 \neq q$. Therefore, the case of $n > 1$ is also not possible.

Conclusion

In conclusion, we have shown that it is not possible to find two matrices, $U$ and $V$, with integer coefficients such that their product is equal to the original matrix $A$, and the determinants of $U$ and $V$ are both greater than 1. This result has important implications for the study of matrix factorisation and the properties of determinants.

Future Work

There are several possible directions for future research on this problem. One possible approach is to consider the case of matrices with non-integer coefficients. Another possible approach is to consider the case of matrices with a specific structure, such as triangular or diagonal matrices.

References

• [1] Hoffman, K., and Kunze, R. (1971). Linear Algebra. Prentice Hall.
• [2] Lang, S. (1987). Linear Algebra. Springer-Verlag.
• [3] Strang, G. (1988). Linear Algebra and Its Applications. Harcourt Brace Jovanovich.

Appendix

The following is a list of the matrices used in this article:

• $A$: a matrix with integer coefficients
• $U$: a matrix with integer coefficients
• $V$: a matrix with integer coefficients

The following is a list of the determinants used in this article:

• $\det(A)$: the determinant of matrix $A$
• $\det(U)$: the determinant of matrix $U$
• $\det(V)$: the determinant of matrix $V$
  Factorising the Determinant: A Q&A Article =====================================================

Introduction

In our previous article, we explored the concept of factorising the determinant, and we showed that it is not possible to find two matrices, $U$ and $V$, with integer coefficients such that their product is equal to the original matrix $A$, and the determinants of $U$ and $V$ are both greater than 1. In this article, we will answer some of the most frequently asked questions about factorising the determinant.

Q: What is the determinant of a matrix?

A: The determinant of a matrix is a scalar value that can be used to determine the invertibility of the matrix, as well as the volume scaling factor of the linear transformation represented by the matrix.

Q: How is the determinant calculated?

A: The determinant of a matrix can be calculated using various methods, including the expansion by minors, the cofactor expansion, or the use of a determinant formula.

Q: What is the relationship between the determinant of a matrix and its inverse?

A: The determinant of a matrix is equal to the product of the determinants of its inverse and the original matrix. In other words, if $A$ is a matrix and $A^{-1}$ is its inverse, then $\det(A) = \det(A^{-1}) \det(A)$.

Q: Can the determinant of a matrix be zero?

A: Yes, the determinant of a matrix can be zero. In fact, if the determinant of a matrix is zero, then the matrix is not invertible.

Q: What is the significance of the determinant in linear algebra?

A: The determinant is a fundamental concept in linear algebra, and it plays a crucial role in many areas of mathematics and science. It is used to determine the invertibility of matrices, to calculate the volume scaling factor of linear transformations, and to solve systems of linear equations.

Q: Can the determinant of a matrix be negative?

A: Yes, the determinant of a matrix can be negative. In fact, the determinant of a matrix can be any real number, positive or negative.

Q: How does the determinant relate to the eigenvalues of a matrix?

A: The determinant of a matrix is equal to the product of its eigenvalues. In other words, if $A$ is a matrix and $\lambda_1, \lambda_2, \ldots, \lambda_n$ are its eigenvalues, then $\det(A) = \lambda_1 \lambda_2 \cdots \lambda_n$.

Q: Can the determinant of a matrix be expressed as a product of two other matrices?

A: Yes, the determinant of a matrix can be expressed as a product of two other matrices. In fact, if $A$ is a matrix and $U$ and $V$ are two other matrices, then $\det(A) = \det(U) \det(V)$ if and only if $A = UV$.

Q: What is the relationship between the determinant of a matrix and its rank?

A: The determinant of a matrix is equal to the product of its rank and the determinant of its reduced row echelon form. In other words, if $A$ is a matrix and $r$ is its rank, then $\det(A) = r \det(A_{\text{rref}})$.

Q: Can the determinant of a matrix be used to determine its rank?

A: Yes, the determinant of a matrix can be used to determine its rank. In fact, if the determinant of a matrix is zero, then its rank is less than the number of rows or columns.

Conclusion

In conclusion, we have answered some of the most frequently asked questions about factorising the determinant. We hope that this article has provided a useful resource for those interested in linear algebra and the properties of determinants.

References

• [1] Hoffman, K., and Kunze, R. (1971). Linear Algebra. Prentice Hall.
• [2] Lang, S. (1987). Linear Algebra. Springer-Verlag.
• [3] Strang, G. (1988). Linear Algebra and Its Applications. Harcourt Brace Jovanovich.

Appendix

The following is a list of the matrices used in this article:

• $A$: a matrix with integer coefficients
• $U$: a matrix with integer coefficients
• $V$: a matrix with integer coefficients

The following is a list of the determinants used in this article:

• $\det(A)$: the determinant of matrix $A$
• $\det(U)$: the determinant of matrix $U$
• $\det(V)$: the determinant of matrix $V$