(ab)^-1 = A^-1 B^-1 Matrix

5 min read Jun 16, 2024

Understanding the Inverse of a Matrix Product: (AB)^-1 = A^-1 B^-1

In linear algebra, understanding the relationship between matrix multiplication and inverses is crucial. A common question arises regarding the inverse of the product of two matrices, specifically (AB)^-1. This article delves into this concept, explaining why (AB)^-1 = A^-1 B^-1.

The Basics: Inverses and Matrix Multiplication

Inverse: The inverse of a matrix, denoted by A^-1, is a matrix that, when multiplied by the original matrix A, results in the identity matrix (I). This means A * A^-1 = A^-1 * A = I.
Matrix Multiplication: Matrix multiplication is not commutative. This means that AB is not necessarily equal to BA.

Proving the Identity: (AB)^-1 = A^-1 B^-1

Let's consider two invertible matrices A and B. We need to show that the product of their inverses, A^-1 B^-1, acts as the inverse of the product AB. To do this, we'll multiply (AB) by A^-1 B^-1 and demonstrate that we obtain the identity matrix:

Start with (AB) * (A^-1 B^-1)
Rearrange using associativity of matrix multiplication: (AB) * (A^-1 B^-1) = A(B * A^-1)B^-1
Utilize the property that A^-1 * A = I: A(B * A^-1)B^-1 = A(I * B^-1)
Use the fact that I is the identity matrix: A(I * B^-1) = A * B^-1
Rearrange again: A * B^-1 = (A * B^-1) * I
Apply the property that B * B^-1 = I: (A * B^-1) * I = (A * B^-1) * (B * B^-1)
Rearrange using associativity: (A * B^-1) * (B * B^-1) = A(B^-1 * B)B^-1
Utilize the property that B^-1 * B = I: A(B^-1 * B)B^-1 = A * I
Use the fact that I is the identity matrix: A * I = A
Finally, apply the property that A * A^-1 = I: A = I * A = (A^-1 * A) * A = A^-1 (A * A) = A^-1 * I = A^-1

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

We've shown that the product of the inverses of A and B is indeed the inverse of the product AB, confirming the identity: (AB)^-1 = A^-1 B^-1.

Important Notes

This identity only holds true if both matrices A and B are invertible.
The order of the inverses is crucial. A^-1 B^-1 is not equal to B^-1 A^-1, unless A and B commute (AB = BA).

Conclusion

The equation (AB)^-1 = A^-1 B^-1 is a fundamental concept in linear algebra, allowing us to efficiently calculate the inverse of a product of invertible matrices. Understanding this principle is essential for solving various problems involving matrices and their transformations.