The Product of Singular Matrices: Understanding Singularity in Matrix Multiplication

In the realm of linear algebra, the concept of singular matrices is fundamental. A singular matrix is a square matrix that does not have an inverse; this is characterized by its determinant being zero. Understanding whether the product of two singular matrices is also singular is crucial for various applications, including solving systems of linear equations, transformations, and more. This article delves into the conditions and properties that ensure the product of two singular matrices remains singular.

Definitions and Background

To lay a foundation, let's revisit the definition of a singular matrix. A square matrix is singular if its determinant is zero, indicating that the matrix does not have an inverse. This non-invertibility means that the matrix cannot be transformed into the identity matrix under elementary row operations. Another important characteristic of a singular matrix is that its null space is non-trivial, implying that there exists a non-zero vector in the null space.

Product of Two Singular Matrices

Now, let's explore the product of two singular matrices, A and B. The critical insight comes from the properties of determinants and the null space of the matrices. Consider the following points:

Determinant Property

The determinant of a product of matrices follows a specific rule: for any two square matrices A and B, the determinant of their product is given by:

det(AB) det(A) × det(B)

Since a singular matrix has a determinant of zero, if A and B are both singular, then:

det(A) 0 and det(B) 0

Applying the determinant property:

det(AB) det(A) × det(B) 0 × 0 0

This implies that AB is also a singular matrix because its determinant is zero.

Null Space Property

Another way to understand this is through the null space. If a matrix B has a non-trivial null space, then there exists a non-zero vector x such that:

Bx 0

When we consider the product AB, if A and B are both singular, the null space of B will influence the null space of AB. Specifically, if Bx 0 for some non-zero vector x, then:

ABx A(0) 0

Thus, AB also has a non-trivial null space, confirming its singularity.

Order and Assumptions

It is important to note that these properties hold true specifically for square matrices. For non-square matrices, the product may not be defined, or the properties of singularity might differ. Additionally, the order of multiplication (i.e., whether we consider AB or BA) does not affect the singularity in the case of square singular matrices because the determinant property and null space conditions are symmetric.

Example and Conclusion

To illustrate, let's consider a simple example:

Let A and B be 2x2 zero matrices:

A [[0, 0], [0, 0]]

B [[0, 0], [0, 0]]

The product:

AB [[0, 0], [0, 0]]

is clearly a singular matrix because:

det(AB) 0 and AB has a non-trivial null space.

In summary, the product of two singular matrices is indeed singular, as the determinant of the product will always be zero, and the null space of the product will be non-trivial if the original matrices are singular.