(ab)^-1=b^-1a^-1 Matrix Proof

4 min read Jun 16, 2024

Proving the Matrix Inverse Property: (AB)^-1 = B^-1A^-1

This article will provide a detailed proof of the important matrix property: (AB)^-1 = B^-1A^-1. This property states that the inverse of the product of two matrices is equal to the product of their inverses, but in reverse order.

Understanding the Concept

Before diving into the proof, it's important to understand the concepts involved:

Matrix Inverse: A matrix A has an inverse, denoted by A^-1, if their product is the identity matrix (I): A * A^-1 = I.
Matrix Multiplication: Matrix multiplication is not commutative. This means AB ≠ BA in general.

Proof using the Identity Matrix

We can prove the property by showing that B^-1A^-1 satisfies the definition of the inverse of AB.

Let's start by multiplying (AB) and (B^-1A^-1):

(AB) * (B^-1A^-1)

Using the associativity of matrix multiplication, we can rearrange the terms:

A(BB^-1)A^-1

Since BB^-1 = I (by the definition of inverse matrices), we get:

AIA^-1

And since the identity matrix (I) acts as a multiplicative identity:

AA^-1

Again, by the definition of inverse matrices, AA^-1 = I. Therefore, we have:

(AB) * (B^-1A^-1) = I

This shows that B^-1A^-1 is the inverse of AB, proving the property:

(AB)^-1 = B^-1A^-1

Importance of the Property

This property is crucial for several reasons:

Simplifying Calculations: It allows us to find the inverse of a product of matrices by inverting each matrix individually and multiplying them in reverse order.
Solving Linear Equations: This property plays a fundamental role in solving systems of linear equations using matrix methods.
Understanding Linear Transformations: The inverse of a matrix represents the inverse of a linear transformation. This property helps us understand how the inverse of a composite transformation is related to the inverses of individual transformations.

Conclusion

The property (AB)^-1 = B^-1A^-1 is a key element in understanding and manipulating matrices. This proof, derived from the fundamental definition of inverse matrices, demonstrates the logical foundation for this property. By understanding this relationship, we gain a deeper understanding of how matrix operations work and how they can be applied in various areas of mathematics, science, and engineering.