To prove the statement: If $A$ is orthogonal, then $A^{-1} = A^T$:

Proof:

1. Definition of Orthogonal Matrix: A matrix $A$ is orthogonal if: $A^T A = I$ where $A^T$ is the transpose of $A$, and $I$ is the identity matrix.

2. Invertibility of Orthogonal Matrices: By the property of matrix inverses, multiplying $A$ by its inverse $A^{-1}$ gives the identity matrix: $A A^{-1} = I$

3. Use the Orthogonality Condition: From the definition of orthogonality: $A^T A = I$ Multiply both sides by $A^{-1}$: $A^{-1} A^T A = A^{-1} I$ Simplify: $A^T = A^{-1}$

4. Conclusion: Therefore, for an orthogonal matrix $A$: $A^{-1} = A^T$

This completes the proof.

Would you like more details or an explanation of any specific part?

Related Questions:

1. What are the properties of orthogonal matrices?
2. How is the determinant of an orthogonal matrix related to its properties?
3. Can a non-square matrix be orthogonal?
4. How is the orthogonality condition used in numerical applications?
5. Why does $A^{-1}$ exist for orthogonal matrices?

Tip:

Always verify orthogonality by checking $A^T A = I$ when working with matrices in computations or proofs.