(AB)⁻¹ = A⁻¹ ⋅ B⁻¹, where A and B are invertible matrices satisfying commutative property with respect to multiplication.

The equation (AB)⁻¹ = A⁻¹ ⋅ B⁻¹ describes the relationship between the inverses of two invertible matrices, A and B. This relationship holds true when A and B commute, meaning that AB = BA.

Understanding Matrix Inverses

In linear algebra, the inverse of a matrix A, denoted as A⁻¹, is a matrix that, when multiplied by A, yields the identity matrix I. This is crucial for solving systems of linear equations and other applications.

Conditions for the Inverse Property

• Invertibility: Both matrices A and B must be invertible.
• Commutativity: The matrices must satisfy the condition AB = BA.

Why This Property Matters

This property is significant because it simplifies calculations involving the product of matrices. When A and B commute, finding the inverse of their product becomes straightforward, allowing for easier manipulation in various mathematical contexts.

Example

Consider two matrices A and B. If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], and if they commute, then:

(AB)⁻¹ = A⁻¹ ⋅ B⁻¹ can be computed directly using their individual inverses.

In summary, the equation (AB)⁻¹ = A⁻¹ ⋅ B⁻¹ is a powerful tool in linear algebra, provided the matrices involved are both invertible and commute with each other.