1. **Problem statement:** Verify that for a matrix $A$ and integer $k \geq 1$, the transpose of the power satisfies $$(A^k)^T = (A^T)^k.$$ 2. **Recall the transpose property:** For any two matrices $X$ and $Y$ of compatible sizes, $$(XY)^T = Y^T X^T.$$ 3. **Base case ($k=1$):** Clearly, $$(A^1)^T = A^T = (A^T)^1,$$ so the property holds for $k=1$. 4. **Inductive step:** Assume the property holds for some $k = n$, i.e., $$(A^n)^T = (A^T)^n.$$ 5. **Show it holds for $k = n+1$: $$(A^{n+1})^T = (A^n A)^T = A^T (A^n)^T$$ by the transpose of a product. 6. **Use the induction hypothesis:** Substitute $$(A^n)^T = (A^T)^n$$ to get $$(A^{n+1})^T = A^T (A^T)^n = (A^T)^{n+1}.$$ 7. **Conclusion:** By mathematical induction, the property holds for all integers $k \geq 1$: $$ (A^k)^T = (A^T)^k.$$ This completes the verification.