A lock has $16$ keys arranged in a $4 \times 4$  array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too (see diagram). Show that no matter what the starting
positions are, it is always possible to open this lock. (Only one key at a time can be switched.)


1. **Representation of the Problem:**
   - We have a $4 \times 4$ array of keys, each of which can be in one of two states: horizontal (H) or vertical (V).
   - The goal is to make all keys vertical (V).
   - When a key is switched, all keys in the same row and column also switch their states.

2. **Mathematical Representation:**
   - Represent each key's state as a binary variable in $\mathbb{F}_2$ (the field with two elements, 0 and 1).
   - Let $x_{ij} \in \mathbb{F}_2$ represent the state of the key in the $i$-th row and $j$-th column, where $0$ represents vertical (V) and $1$ represents horizontal (H).

3. **Operation Matrix:**
   - Define a $16 \times 16$ matrix $A$ over $\mathbb{F}_2$ where each row corresponds to an operation (switching a key) and each column corresponds to a key's state.
   - The entry $A_{kl}$ is $1$ if the $k$-th operation affects the $l$-th key, and $0$ otherwise.

4. **Matrix Construction:**
   - Each operation affects all keys in the same row and column as the key being switched.
   - For example, switching the key at position $(i, j)$ affects all keys in the $i$-th row and $j$-th column.

5. **Determinant and Linear Independence:**
   - To show that it is always possible to open the lock, we need to show that the matrix $A$ is invertible over $\mathbb{F}_2$.
   - This is equivalent to showing that the determinant of $A$ is non-zero in $\mathbb{F}_2$.

6. **Determinant Calculation:**
   - Construct the matrix $A$ explicitly and compute its determinant.
   - If the determinant is non-zero, then the matrix is invertible, and any initial configuration can be transformed to the desired configuration (all keys vertical).

7. **Generalization for Even $n$:**
   - The construction holds for any even $n \times n$ array.
   - For odd $n > 1$, the statement is false because picking two rows and operating on all elements gives the identity, which contradicts linear independence.

8. **Conclusion:**
   - Since the determinant of the $16 \times 16$ matrix $A$ is non-zero in $\mathbb{F}_2$, the matrix is invertible.
   - Therefore, it is always possible to open the lock regardless of the starting positions of the keys.

\[
\boxed{\text{True}}
\]