(a^-1)^t = (a^t)^-1 Matrix Proof

Jasonbradley

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Jun 16, 2024

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Proving the Matrix Identity: (A⁻¹)^t = (A^t)⁻¹

This article aims to prove the matrix identity: (A⁻¹)^t = (A^t)⁻¹, where:

• A is an invertible matrix
• A⁻¹ is the inverse of matrix A
• A^t is the transpose of matrix A

Understanding the Identity

The identity essentially states that the transpose of the inverse of a matrix is equal to the inverse of its transpose. This property holds true for any invertible matrix and has significant applications in linear algebra and matrix theory.

Proof:

To prove this identity, we will use the following two properties:

1. (AB)^t = B^t A^t: The transpose of the product of two matrices is equal to the product of their transposes in reverse order.
2. (A⁻¹)(A) = (A)(A⁻¹)= I: The product of a matrix and its inverse results in the identity matrix (I).

Steps:

1. Start with the left-hand side of the identity: (A⁻¹)^t
2. Multiply it by A^t and its inverse (A^t)⁻¹: (A⁻¹)^t * (A^t) * (A^t)⁻¹
3. Rearrange the terms: (A⁻¹)^t * A^t * (A^t)⁻¹ = ((A⁻¹)^t * A^t) * (A^t)⁻¹
4. Apply the first property mentioned above: ((A⁻¹)^t * A^t) * (A^t)⁻¹ = (A^t * (A⁻¹)^t) * (A^t)⁻¹
5. Apply the second property mentioned above: (A^t * (A⁻¹)^t) * (A^t)⁻¹ = I * (A^t)⁻¹ = (A^t)⁻¹

Therefore, we have shown that (A⁻¹)^t = (A^t)⁻¹

Conclusion:

This proof demonstrates the validity of the matrix identity (A⁻¹)^t = (A^t)⁻¹ using basic matrix properties. This identity has practical applications in various areas like solving linear systems, performing matrix operations, and understanding matrix transformations.