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.