Is The Product Of Two Invertible Matrices Invertible

Is the Product of Two Invertible Matrices Invertible? A Deep Dive into Linear Algebra

The question of whether the product of two invertible matrices is itself invertible is a fundamental concept in linear algebra. The short answer is a resounding yes, and understanding why unveils crucial properties of matrices and their applications. This article will delve into a detailed explanation, exploring the underlying theory, providing rigorous proofs, and illustrating the concept with examples. We'll also touch upon the implications of this property in various fields.

Understanding Invertible Matrices

Before diving into the product, let's solidify our understanding of invertible matrices. A square matrix (same number of rows and columns) is considered invertible (also known as nonsingular or nondegenerate) if there exists another matrix, called its inverse, that, when multiplied with the original matrix, yields the identity matrix. The identity matrix, denoted by I, is a square matrix with 1s along the main diagonal and 0s elsewhere.

Formally, for a square matrix A, it's invertible if there exists a matrix A⁻¹ such that:

A * A⁻¹ = A⁻¹ * A = I

A matrix that doesn't have an inverse is called singular or non-invertible. Singular matrices have a determinant of zero, while invertible matrices have a non-zero determinant. This determinant property provides a quick way to check for invertibility.

Proving the Invertibility of the Product

Now, let's tackle the core question: If A and B are invertible square matrices of the same size, is their product AB also invertible? The answer is yes, and we can prove this using the definition of invertibility.

Theorem: If A and B are invertible n x n matrices, then their product AB is also invertible, and (AB)⁻¹ = B⁻¹A⁻¹.

Proof: To prove that AB is invertible, we need to show that there exists a matrix that, when multiplied by AB, results in the identity matrix I. Let's consider the product of AB and B⁻¹A⁻¹:

(AB)(B⁻¹A⁻¹) = A(BB⁻¹)A⁻¹ = A(I)A⁻¹ = AIA⁻¹ = AA⁻¹ = I

Similarly, we can show that:

(B⁻¹A⁻¹)(AB) = B⁻¹(A⁻¹A)B = B⁻¹(I)B = B⁻¹IB = B⁻¹B = I

Since both (AB)(B⁻¹A⁻¹) = I and (B⁻¹A⁻¹)(AB) = I, we have successfully demonstrated that B⁻¹A⁻¹ is the inverse of AB. Therefore, the product AB is invertible, and its inverse is the product of the inverses in reverse order. This completes the proof.

This proof highlights a crucial point: the order of multiplication matters when dealing with matrix inverses. The inverse of the product is not simply the product of the inverses; it's the product of the inverses in the reversed order.

Implications and Applications

The invertibility of the product of invertible matrices has far-reaching consequences across numerous fields:

• Linear Transformations: Invertible matrices represent invertible linear transformations. The product of two invertible matrices represents the composition of two invertible linear transformations. Our theorem guarantees that the composition of two invertible transformations is also invertible. This is fundamental in understanding how linear transformations affect vector spaces.

• Solving Systems of Linear Equations: Invertible matrices are central to solving systems of linear equations. If a system of equations can be represented in matrix form as Ax = b, where A is an invertible matrix, then the solution is given by x = A⁻¹b. If we have a sequence of linear transformations represented by matrices A and B, the combined effect is represented by AB. The invertibility of AB ensures we can solve the combined system.

• Computer Graphics: In computer graphics, matrices are heavily used for transformations such as rotation, scaling, and translation. Combining these transformations involves matrix multiplication. The fact that the product of invertible matrices is invertible ensures that complex transformations can be reversed, allowing for manipulations and animations.

• Cryptography: Invertible matrices play a crucial role in certain cryptographic systems. The ability to easily invert matrices is essential for encryption and decryption processes. The properties of invertible matrix products are fundamental for secure and efficient cryptographic algorithms.

• Signal Processing: Matrix operations are fundamental in signal processing for tasks like filtering and compression. Invertibility of matrices is essential to ensure that signals can be processed and reconstructed without loss of information. The composition of multiple signal processing operations, often represented by matrix multiplication, relies on the invertibility of the product.

Examples

Let's illustrate the concept with a couple of examples:

Example 1:

Let's consider two 2x2 invertible matrices:

A = [[2, 1], [1, 1]]

B = [[1, -1], [0, 2]]

The inverses are:

A⁻¹ = [[1, -1], [-1, 2]]

B⁻¹ = [[1, 1/2], [0, 1/2]]

Their product is:

AB = [[2, 0], [1, 1]]

And the inverse of the product, calculated using the formula (AB)⁻¹ = B⁻¹A⁻¹, is:

(AB)⁻¹ = [[1, 1/2], [0, 1/2]] * [[1, -1], [-1, 2]] = [[1/2, 0], [-1/2, 1]]

Let's verify:

(AB)(AB)⁻¹ = [[2, 0], [1, 1]] * [[1/2, 0], [-1/2, 1]] = [[1, 0], [0, 1]] = I

Example 2 (Illustrating Non-Commutativity):

This example further emphasizes the importance of the order of operations when finding the inverse of a product. Let's swap the order of matrices A and B from Example 1:

BA = [[1, -1], [0, 2]] * [[2, 1], [1, 1]] = [[1, 0], [2, 2]]

(BA)⁻¹ = A⁻¹B⁻¹ = [[1, -1], [-1, 2]] * [[1, 1/2], [0, 1/2]] = [[1, -1], [-1, 3/2]]

Let's verify:

(BA)(BA)⁻¹ = [[1, 0], [2, 2]] * [[1, -1], [-1, 3/2]] = [[1, 0], [0, 1]] = I

Notice how (AB)⁻¹ ≠ (BA)⁻¹. This underscores the non-commutative nature of matrix multiplication.

Conclusion

The invertibility of the product of two invertible matrices is a cornerstone of linear algebra. Its proof, based on the definition of matrix inverses, is elegant and straightforward. The implications of this property are far-reaching, impacting diverse fields that rely heavily on linear transformations and matrix operations. Understanding this fundamental concept is critical for anyone working with matrices and their applications. The examples provided illustrate not only the validity of the theorem but also the non-commutative nature of matrix multiplication regarding inverses. This deep dive hopefully provides a solid understanding of this important mathematical concept.