Proving (A⁻¹)¹ = A for Matrices
In linear algebra, 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). The identity matrix is a square matrix with 1's on the diagonal and 0's elsewhere.
One important property of matrix inverses is that the inverse of the inverse of a matrix is the original matrix itself. In other words, (A⁻¹)¹ = A. Let's explore the proof for this property.
Proof:

Definition of Inverse: We know that A⁻¹ is the inverse of A, which means:
A * A⁻¹ = I and A⁻¹ * A = I

Inverse of A⁻¹: We are looking for (A⁻¹)¹, which is the inverse of A⁻¹. This means:
(A⁻¹)¹ * A⁻¹ = I

Multiplying by A: Multiply both sides of the equation in step 2 by A:
(A⁻¹)¹ * A⁻¹ * A = I * A

Associative Property: Using the associative property of matrix multiplication, we can rearrange the terms:
(A⁻¹)¹ * (A⁻¹ * A) = A

Definition of Inverse: From step 1, we know that A⁻¹ * A = I. Substituting this into the equation:
(A⁻¹)¹ * I = A

Identity Property: The identity matrix (I) acts as a multiplicative identity, meaning that multiplying any matrix by I leaves it unchanged. Therefore:
(A⁻¹)¹ = A
Conclusion:
We have successfully proven that (A⁻¹)¹ = A for any invertible matrix A. This property demonstrates the cyclical nature of matrix inverses, emphasizing that the inverse of the inverse returns the original matrix.