In matrix arithmetic, addition and subtraction of matrices with integer entries follow straightforward rules as long as the matrices have the same dimensions. Let's go over these operations in detail.

Matrix Addition

To add two matrices, say $A$ and $B$, they must have the same dimensions. Each element in the resulting matrix $C = A + B$ is obtained by adding the corresponding elements of matrices $A$ and $B$.

For example, if: $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$ then $A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$

Matrix Subtraction

For matrix subtraction, the matrices must again have the same dimensions. Each element in the resulting matrix $D = A - B$ is obtained by subtracting the corresponding elements of matrix $B$ from matrix $A$.

Continuing from the example above: $A - B = \begin{bmatrix} 1-5 & 2-6 \\ 3-7 & 4-8 \end{bmatrix} = \begin{bmatrix} -4 & -4 \\ -4 & -4 \end{bmatrix}$

Properties of Matrix Addition and Subtraction

1. Commutativity of Addition: $A + B = B + A$
2. Associativity of Addition: $(A + B) + C = A + (B + C)$
3. Additive Identity: For a matrix $A$, there exists a zero matrix $O$ such that $A + O = A$.
4. Additive Inverse: For any matrix $A$, there exists a matrix $-A$ such that $A + (-A) = O$, where $O$ is the zero matrix.

Would you like more details, or have any questions?

Here are some related questions to explore:

1. What are examples of scalar multiplication in matrices?
2. How do you determine if matrices can be added or subtracted?
3. What is matrix multiplication, and how does it differ from addition?
4. Can you provide examples of real-life applications of matrix operations?
5. How is matrix addition different for matrices with fractional entries?

Tip: Ensure matrices have the same dimensions before performing addition or subtraction.