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

Jasonbradley

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Jun 16, 2024

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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:

1. Definition of Inverse: We know that A⁻¹ is the inverse of A, which means:

   A * A⁻¹ = I and A⁻¹ * A = I

2. 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

3. Multiplying by A: Multiply both sides of the equation in step 2 by A:

   (A⁻¹)-¹ * A⁻¹ * A = I * A

4. Associative Property: Using the associative property of matrix multiplication, we can rearrange the terms:

   (A⁻¹)-¹ * (A⁻¹ * A) = A

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

   (A⁻¹)-¹ * I = A

6. 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.