Question
If \(A\) is a square matrix, prove using mathematical induction that \[ (A^T)^n = (A^n)^T \quad \forall n \in \mathbb{N}. \]
Solution (Mathematical Induction)
Step 1: Base Case (n = 1)
\[ (A^T)^1 = A^T \quad \text{and} \quad (A^1)^T = A^T \] ✔ True for \(n=1\)Step 2: Assume for \(n = k\)
\[ (A^T)^k = (A^k)^T \]Step 3: Prove for \(n = k+1\)
\[ (A^T)^{k+1} = (A^T)^k \cdot A^T \] Using assumption: \[ = (A^k)^T \cdot A^T \]Step 4: Use Property
\[ (XY)^T = Y^T X^T \] \[ (A^k)^T A^T = (A \cdot A^k)^T \]Step 5: Simplify
\[ (A \cdot A^k)^T = (A^{k+1})^T \] \[ \Rightarrow (A^T)^{k+1} = (A^{k+1})^T \]Step 6: Conclusion
✔ True for \(k+1\) \[ \Rightarrow (A^T)^n = (A^n)^T \]Final Result
\[
(A^T)^n = (A^n)^T
\]
Hence proved.