Proof The Inverse Of A Product Of Matrices DE - Linear Algebra

Let $D$ and $E$ be two square matrices of the same size such that $\operatorname{det}(D) \neq 0$ and $\operatorname{det}(E) \neq 0$. Prove that the product $DE$ has an inverse and $E^{-1} D^{-1}=(D E)^{-1}$.

In the realm of linear algebra, matrices hold a fundamental role, serving as powerful tools for representing and manipulating linear transformations. Among the various operations that can be performed on matrices, multiplication stands out as a cornerstone, enabling the composition of transformations and the construction of complex systems. However, the properties of matrix multiplication extend beyond mere composition, encompassing concepts such as invertibility and the existence of inverse matrices. In this article, we delve into the intricacies of matrix inverses, specifically focusing on the product of two matrices and its inverse. Our primary objective is to rigorously prove that if $D$ and $E$ are two square matrices of the same size with nonzero determinants, then their product $DE$ has an inverse, and this inverse is given by $E^{-1}D^{-1}$. This fundamental result provides crucial insights into the structure of matrix inverses and their behavior under multiplication, paving the way for numerous applications in diverse fields.

Before embarking on the proof, it is essential to establish the necessary prerequisites and definitions. We assume familiarity with basic matrix operations, including matrix multiplication, determinants, and the concept of an inverse matrix. Recall that a square matrix $A$ is said to be invertible if there exists a matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix. The matrix $A^{-1}$ is called the inverse of $A$. Furthermore, the determinant of a matrix, denoted by $\operatorname{det}(A)$, is a scalar value that provides valuable information about the matrix's properties, including its invertibility. A fundamental theorem states that a square matrix $A$ is invertible if and only if its determinant is nonzero, i.e., $\operatorname{det}(A) \neq 0$. This theorem serves as a cornerstone for our proof, allowing us to connect the invertibility of a matrix to its determinant. In addition to these fundamental concepts, we will also utilize the property that the determinant of a product of matrices is equal to the product of their determinants, i.e., $\operatorname{det}(AB) = \operatorname{det}(A) \operatorname{det}(B)$. This property will be instrumental in establishing the invertibility of the product $DE$.

Theorem: Let $D$ and $E$ be two square matrices of the same size such that $\operatorname{det}(D) \neq 0$ and $\operatorname{det}(E) \neq 0$. Then the product $DE$ has an inverse, and $E^{-1}D^{-1} = (DE)^{-1}$.

Proof:

Our proof unfolds in two key steps. First, we demonstrate that the product $DE$ is invertible. Second, we establish that $E^{-1}D^{-1}$ is indeed the inverse of $DE$.

Step 1: Proving the Invertibility of DE

To establish the invertibility of $DE$, we leverage the fundamental theorem that connects invertibility to determinants. We need to show that $\operatorname{det}(DE) \neq 0$. Using the property that the determinant of a product is the product of determinants, we have:

$$\operatorname{det}(DE) = \operatorname{det}(D) \operatorname{det}(E)$$

Since we are given that $\operatorname{det}(D) \neq 0$ and $\operatorname{det}(E) \neq 0$, their product is also nonzero:

$$\operatorname{det}(D) \operatorname{det}(E) \neq 0$$

Therefore,

$$\operatorname{det}(DE) \neq 0$$

By the theorem mentioned earlier, this implies that the matrix $DE$ is invertible.

Step 2: Verifying the Inverse

Now that we have established the invertibility of $DE$, we proceed to verify that $E^{-1}D^{-1}$ is its inverse. To do this, we need to show that $(DE)(E^{-1}D^{-1}) = I$ and $(E^{-1}D^{-1})(DE) = I$, where $I$ is the identity matrix.

Let's first consider the product $(DE)(E^{-1}D^{-1})$. Using the associative property of matrix multiplication, we can rewrite this as:

$$(DE)(E^{-1}D^{-1}) = D(EE^{-1})D^{-1}$$

Since $EE^{-1} = I$, we have:

$$D(EE^{-1})D^{-1} = DID^{-1}$$

And since $DI = D$, we get:

$$DID^{-1} = DD^{-1}$$

Finally, since $DD^{-1} = I$, we conclude that:

$$(DE)(E^{-1}D^{-1}) = I$$

Now, let's consider the product $(E^{-1}D^{-1})(DE)$. Again, using the associative property, we have:

$$(E^{-1}D^{-1})(DE) = E^{-1}(D^{-1}D)E$$

Since $D^{-1}D = I$, we get:

$$E^{-1}(D^{-1}D)E = E^{-1}IE$$

And since $IE = E$, we have:

$$E^{-1}IE = E^{-1}E$$

Finally, since $E^{-1}E = I$, we conclude that:

$$(E^{-1}D^{-1})(DE) = I$$

Thus, we have shown that both $(DE)(E^{-1}D^{-1}) = I$ and $(E^{-1}D^{-1})(DE) = I$. This confirms that $E^{-1}D^{-1}$ is indeed the inverse of $DE$, i.e., $(DE)^{-1} = E^{-1}D^{-1}$.

In this article, we have successfully proven a fundamental theorem in linear algebra: if $D$ and $E$ are two square matrices of the same size with nonzero determinants, then their product $DE$ has an inverse, and this inverse is given by $E^{-1}D^{-1}$. This result provides valuable insights into the behavior of matrix inverses under multiplication and has far-reaching implications in various fields, including engineering, physics, and computer science. The ability to invert a product of matrices is crucial for solving linear systems of equations, performing transformations, and analyzing complex systems. The theorem we have proven provides a powerful tool for understanding and manipulating matrices, furthering our comprehension of linear algebra and its applications. Understanding the inverse of matrix products is key to solving complex matrix equations and simplifying linear transformations. This property is essential in various applications where matrix operations are used extensively.

The theorem we have proven has numerous applications and implications in various fields. In linear algebra itself, it provides a powerful tool for manipulating matrices and solving linear systems of equations. For example, consider a system of linear equations represented by the matrix equation $Ax = b$, where $A$ is a square matrix, $x$ is the vector of unknowns, and $b$ is the constant vector. If $A$ is invertible, we can multiply both sides of the equation by $A^{-1}$ to obtain $x = A^{-1}b$, which provides the solution to the system. If $A$ can be expressed as a product of matrices, say $A = DE$, then the inverse of $A$ can be found using the theorem we have proven: $A^{-1} = (DE)^{-1} = E^{-1}D^{-1}$. This can simplify the process of finding the inverse and solving the system of equations. Moreover, the theorem has significant implications in areas such as computer graphics and robotics, where matrices are used to represent transformations such as rotations, translations, and scaling. The inverse of a transformation matrix represents the reverse transformation, and the theorem allows us to find the inverse of a composite transformation by multiplying the inverses of the individual transformations in reverse order. This is crucial for tasks such as undoing a series of transformations or finding the transformation that maps one object to another. In addition, the theorem has applications in quantum mechanics, where matrices are used to represent quantum operators. The inverse of an operator represents the reverse operation, and the theorem allows us to find the inverse of a composite operator. The inverse of matrix products is crucial in solving linear systems and inverting transformations, which are fundamental in engineering and physics. This principle allows for efficient computation and analysis in various applications.

This exploration into the inverse of a product of matrices opens doors to further investigation and understanding of related concepts. One natural extension is to consider the invertibility of the sum of matrices. While the inverse of a product has a neat formula, the inverse of a sum does not have a general closed-form expression. However, specific cases and bounds on the invertibility of sums of matrices have been studied extensively. Another avenue for further exploration is the concept of generalized inverses, which provide a notion of an inverse even for non-square or singular matrices. These generalized inverses, such as the Moore-Penrose pseudoinverse, have applications in solving least-squares problems and dealing with ill-conditioned systems. Furthermore, the properties of matrix inverses are closely related to the eigenvalues and eigenvectors of the matrix. Understanding the eigenvalues and eigenvectors can provide insights into the invertibility and stability of linear systems. Delving deeper into matrix inverses leads to advanced topics in linear algebra, including generalized inverses and eigenvalue analysis. These concepts are vital for solving complex problems in optimization and control theory.

In conclusion, we have rigorously proven that for square matrices $D$ and $E$ of the same size, with nonzero determinants, the product $DE$ is invertible, and its inverse is given by $(DE)^{-1} = E^{-1}D^{-1}$. This fundamental result showcases the interplay between matrix multiplication and inversion, offering a cornerstone for various applications across mathematics, engineering, and physics. The ability to decompose complex transformations or systems into simpler components, find the inverses of those components, and then recombine them in reverse order to obtain the inverse of the whole, is a powerful technique. This theorem underscores the importance of understanding matrix inverses and their properties, paving the way for solving linear systems, analyzing transformations, and tackling a wide range of problems in diverse fields. This comprehensive exploration of matrix product inverses provides a solid foundation for advanced linear algebra studies. The ability to apply this theorem significantly enhances problem-solving skills in various scientific and engineering domains.