Show That If Ab Is Invertible So Is B

Showing that if ab is invertible, then b is invertible: A Deep Dive into Linear Algebra

This article explores the fundamental concept in linear algebra that proves if the product of two matrices, ab, is invertible, then the matrix b itself must also be invertible. We'll delve into the proof, examine its implications, and address common misconceptions. Understanding this theorem is crucial for grasping matrix operations and their applications in various fields like computer graphics, machine learning, and physics. We will navigate through the concepts with clarity and precision, ensuring a comprehensive understanding for all readers.

Introduction: The Invertibility of Matrices

Before we jump into the proof, let's establish a solid foundation. A square matrix A is considered invertible (also called nonsingular or nondegenerate) if there exists a matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I, where I represents the identity matrix. The matrix A⁻¹ is called the inverse of A. Not all square matrices are invertible; those that are not are called singular or degenerate. A singular matrix has a determinant of zero, while an invertible matrix has a non-zero determinant. This determinant condition provides a quick test for invertibility.

The Proof: Demonstrating b's Invertibility

Our objective is to demonstrate that if the product of two square matrices, ab, is invertible, then matrix b must also be invertible. Let's proceed step-by-step:

1. Given: We are given that the matrix product ab is invertible. This means there exists a matrix (ab)⁻¹ such that (ab)(ab)⁻¹ = (ab)⁻¹(ab) = I.

2. Finding a Candidate for b⁻¹: Our goal is to find a matrix that, when multiplied by b, yields the identity matrix. Let's consider the matrix (ab)⁻¹a. We'll show this is the inverse of b.

3. Left Multiplication: Let's multiply b from the left by (ab)⁻¹a:

   (ab)⁻¹ab = (ab)⁻¹(ab) = I*

   This equation demonstrates that multiplying b by (ab)⁻¹a from the left results in the identity matrix.

4. Right Multiplication: Now, let's multiply b from the right by (ab)⁻¹a:

   This step is not strictly necessary to prove the invertibility of b. The left multiplication alone is sufficient. However, to show that (ab)⁻¹a is truly the inverse of b, we need to show that the product is the identity matrix when multiplying from both the left and the right. The proof is slightly more complicated in this direction, so we will address it further below.

5. Conclusion: Since we have found a matrix, (ab)⁻¹a, which satisfies the condition b((ab)⁻¹a*) = I* (and as we'll see, (ab)⁻¹ab = I), we conclude that b is invertible, and its inverse is (ab)⁻¹a.

A Deeper Look at Right Multiplication and the Full Inverse Proof

While the left multiplication alone sufficiently proves the invertibility of b, demonstrating the right multiplication provides a more complete picture and solidifies the understanding of the inverse relationship. To prove b(*(ab)⁻¹a) = I requires showing (ab)⁻¹(ab) = I.

The key here lies in the associativity of matrix multiplication. The expression (ab)⁻¹(ab) is unambiguous, and the result is the identity matrix. This is a fundamental property of matrix inverses and should be considered when thinking about matrix operations.

Let's consider matrix multiplication of the form X(YZ). Associativity states that X(YZ) = (XY)Z. Therefore, the order of multiplication doesn't affect the result as long as the order of individual matrices are respected.

Because (ab)(ab)⁻¹ = I, it inherently follows that b(*(ab)⁻¹a) = I and therefore (ab)⁻¹a is indeed the inverse of b.

Implications and Applications

This theorem has significant implications within linear algebra and its applications:

• System of Linear Equations: In the context of systems of linear equations, represented as Ax = b, the invertibility of A guarantees a unique solution. If A is a product of matrices, say ab, this theorem ensures that if the overall system has a unique solution (meaning ab is invertible), then the sub-system represented by b also has specific properties that allow for unique solutions to be found.

• Matrix Decomposition: Many matrix decomposition techniques, like LU decomposition or QR decomposition, rely on the invertibility of certain sub-matrices. This theorem helps analyze the conditions under which these decompositions are possible.

• Linear Transformations: Matrices represent linear transformations. If the composition of two linear transformations (represented by matrix multiplication ab) is invertible, then the individual transformation b must also be invertible. This has significant implications in fields like computer graphics, where transformations are fundamental.

• Error Analysis: In numerical computations involving matrices, understanding invertibility is crucial for identifying potential sources of error and instability. This theorem provides insight into the propagation of errors when dealing with matrix products.

Common Misconceptions

• Assuming a is also invertible: The theorem does not state that a is invertible. It only demonstrates the invertibility of b. a could be singular (non-invertible) and this theorem still holds true.

• Confusing the order of multiplication: The order of multiplication in matrices is crucial. If ba is invertible, it doesn't necessarily mean b is invertible (and similarly, a being invertible doesn't imply ba being invertible). The theorem specifically addresses the invertibility of ab.

• Generalizing to non-square matrices: This theorem applies only to square matrices. The concept of invertibility, as discussed, is specific to square matrices. For non-square matrices, the concept of invertibility must be revised (and is usually associated with right and left pseudo-inverses).

Frequently Asked Questions (FAQ)

Q1: Why is the determinant condition important in this context?

A1: The determinant of a matrix being non-zero is a necessary and sufficient condition for invertibility. If det(ab) ≠ 0, then it doesn't directly imply det(b) ≠ 0. However, the fact that (ab)⁻¹ exists implies that det(ab) ≠ 0, and our proof shows that this subsequently means that b must also be invertible and therefore det(b) ≠ 0.

Q2: Can you provide a simple example illustrating this theorem?

A2: Consider the matrices:

a = [[2, 0], [0, 1]] b = [[1, 1], [1, 0]]

Then ab = [[2, 2], [1, 0]].

Calculating the determinant of ab (det(ab) = -2), we find that ab is invertible. The inverse of b is [[0, 1], [1, -1]] and as we have established (ab)⁻¹a = b⁻¹

Q3: What happens if ab is invertible, but ba is not?

A3: This is perfectly possible. Matrix multiplication is not commutative, meaning ab ≠ ba in general. The invertibility of one does not guarantee the invertibility of the other.

Conclusion

We've rigorously proven that if the product of two square matrices, ab, is invertible, then the matrix b must also be invertible. This seemingly simple theorem holds profound implications across diverse fields that utilize linear algebra. Understanding this concept is vital for developing a strong foundation in linear algebra and its wide-ranging applications. Remember to carefully consider the nuances of matrix multiplication and invertibility to avoid common misconceptions and to fully appreciate the power of this fundamental theorem. The concepts presented here offer a solid base for further exploration into more advanced matrix operations and their applications within various mathematical and computational disciplines.