The expression $A \cdot A = A$, assuming $A$ represents a logical or Boolean variable. Let’s analyze the problem systematically based on Boolean algebra:

Case 1: If $A$ is a Boolean value (e.g., $0$ or $1$):

1. $A \cdot A$ means $A$ AND $A$, which simplifies to $A$ itself. This follows the idempotent law in Boolean algebra.
2. Extending this logic, $A \cdot A \cdot A \cdot A = A$.

Case 2: If $A$ is a matrix:

1. Matrix multiplication is associative, so $A \cdot A \cdot A \cdot A$ implies repeated multiplication of matrix $A$ by itself four times, resulting in $A^4$.

Provided Answer Choices:

Based on the context, this question likely refers to Boolean algebra. Thus:

• $A \cdot A \cdot A \cdot A = A$.
• The correct answer matches C. $A$ $$$$.

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Related Questions:

1. What is the idempotent law in Boolean algebra?
2. How does $A \cdot A = A$ apply in logic circuits?
3. Can this expression apply to non-Boolean scenarios, such as matrices or real numbers?
4. What are the properties of the AND operation in Boolean algebra?
5. How does this concept relate to the laws of sets (e.g., intersections)?

Tip:

In Boolean algebra, remember that $A \cdot A = A$ and $A + A = A$ are fundamental idempotent properties.