## Understanding the (A + B)^2 Matrix

In linear algebra, the square of a matrix sum, represented as **(A + B)²**, is a fundamental operation. It finds applications in various fields, including physics, engineering, and computer science. This article aims to break down the concept and provide a comprehensive understanding of the (A + B)² matrix.

### Defining the Operation

The operation **(A + B)²** is defined as the product of the matrix sum **(A + B)** with itself. It's important to note that matrix multiplication is **not commutative**, meaning **AB ≠ BA** in general. Therefore, we must carefully consider the order of operations.

### Expanding the Expression

To understand the operation, we expand the expression:
**(A + B)² = (A + B)(A + B)**

Using the distributive property, we get:
**(A + B)² = AA + AB + BA + BB**

Finally, we simplify the expression using matrix multiplication:
**(A + B)² = A² + AB + BA + B²**

### Key Points to Remember

**Commutativity:**The order of multiplication matters in matrix algebra.**AB**is generally not equal to**BA**.**Matrix Dimensions:**For matrix addition and multiplication to be valid, the matrices must have compatible dimensions. In this case,**A**and**B**must have the same dimensions.**Simplification:**The expression**(A + B)²**is generally not equal to**A² + 2AB + B²**. This is because matrix multiplication is not commutative, so**AB**and**BA**are not necessarily equal.

### Example

Let's consider two matrices:

**A = [[1, 2], [3, 4]]**
**B = [[5, 6], [7, 8]]**

To calculate (A + B)², we first compute the sum:
**A + B = [[6, 8], [10, 12]]**

Next, we multiply the sum by itself:
**(A + B)² = [[6, 8], [10, 12]] [[6, 8], [10, 12]]**

Performing the matrix multiplication, we get:
**(A + B)² = [[100, 128], [160, 208]]**

### Conclusion

Understanding the (A + B)² matrix operation requires grasping the principles of matrix addition, multiplication, and the non-commutative nature of matrix multiplication. By applying the distributive property and performing matrix multiplication, we can effectively compute the square of a matrix sum. This operation finds applications in various fields, contributing to the versatility and power of linear algebra.