A.   You are given the set *M*(2*Z*) of 2 × 2 matrices with even integer entries. The notation 2*Z* denotes the set of even integers as follows:
*M*(2*Z*) = {[ *a b c d* ]  | *a*, *b*, *c*, and *d* ∈ 2*Z* }
Recall that matrix addition and multiplication are defined as follows for 2 × 2 matrices:
[ *a b c d* ] + [ *w x y  z* ] = [ *a* + *w*    *b* + *x*  *c* + *y*     *d* + *z* ]
[ *a b c d* ] * [ *w x y  z* ] = [ *aw* + *by*    *ax* + *bz*   *cw* + *dy*     *cx* + *dz* ]
The set**** *M*(2*Z*) with the operations of matrix addition and multiplication is a ring.
1.  Using a counterexample, explain why *M*(2*Z*) is not an integral domain. 

### Integral Domain Definition
An integral domain is a type of ring with two key properties:

1. **No Zero Divisors:** If \( A \) and \( B \) are elements of the ring, and \( A \cdot B = 0 \), then either \( A = 0 \) or \( B = 0 \).
2. **Commutative Multiplication:** The multiplication operation in the ring is commutative, i.e., \( A \cdot B = B \cdot A \).

To show that \( M(2\mathbb{Z}) \) is not an integral domain, we need to provide a counterexample demonstrating that there exist non-zero matrices \( A \) and \( B \) in \( M(2\mathbb{Z}) \) such that \( A \cdot B = 0 \).

### Counterexample
Consider the following matrices \( A \) and \( B \) in \( M(2\mathbb{Z}) \):

\[
A = \begin{bmatrix} 2 & 0 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 0 \\ 0 & 2 \end{bmatrix}
\]

Both matrices \( A \) and \( B \) are non-zero, and all their entries are even integers, so \( A, B \in M(2\mathbb{Z}) \).

Now, let's compute the product \( A \cdot B \):

\[
A \cdot B = \begin{bmatrix} 2 & 0 \\ 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 & 0 \\ 0 & 2 \end{bmatrix} = \begin{bmatrix} (2 \times 0) + (0 \times 0) & (2 \times 0) + (0 \times 2) \\ (0 \times 0) + (0 \times 0) & (0 \times 0) + (0 \times 2) \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
\]

The resulting matrix is the zero matrix, \( 0 \), in \( M(2\mathbb{Z}) \).

### Conclusion
We have found non-zero matrices \( A \) and \( B \) such that \( A \cdot B = 0 \). This means that \( M(2\mathbb{Z}) \) has zero divisors and thus **is not** an integral domain.

Explore why M(2Z), the set of 2x2 matrices with even integer entries, is not an integral domain using a counterexample. Understand the concepts of ring theory, matrix algebra, and integral domains with detailed explanations and computations. Suitable for advanced undergraduate level study.

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