## Understanding the Square of a Sum for Matrices

In linear algebra, we often encounter expressions involving matrices, and understanding how to manipulate them is crucial. One common expression is the square of a sum of matrices, represented as **(A + B)²**.

Let's break down this concept and explore its properties.

### Defining the Square of a Sum for Matrices

For matrices A and B with compatible dimensions, the square of their sum (A + B)² is defined as:

**(A + B)² = (A + B) * (A + B)**

This means we multiply the sum of the matrices by itself.

### Expanding the Expression

To understand how to perform this operation, we can expand the expression using the distributive property:

**(A + B)² = (A + B) * (A + B) = A * (A + B) + B * (A + B)**

Applying the distributive property again, we get:

**(A + B)² = A * A + A * B + B * A + B * B**

This can be written more compactly as:

**(A + B)² = A² + AB + BA + B²**

### Important Considerations

It is essential to note that matrix multiplication is **not commutative**. This means that in general, **AB ≠ BA**. Therefore, the terms **AB** and **BA** are not necessarily equal.

This non-commutativity distinguishes the expansion of (A + B)² for matrices from the familiar algebraic expansion of (a + b)² for scalars.

### Example

Let's consider a simple example to illustrate the process:

Suppose we have:

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

To calculate (A + B)², we first calculate A + B:

**A + B = [1 2; 3 4] + [5 6; 7 8] = [6 8; 10 12]**

Then, we multiply (A + B) by itself:

**(A + B)² = [6 8; 10 12] * [6 8; 10 12] = [116 152; 220 292]**

Alternatively, we can use the expanded formula:

**(A + B)² = A² + AB + BA + B²**

**A² = [1 2; 3 4] * [1 2; 3 4] = [7 10; 15 22]****AB = [1 2; 3 4] * [5 6; 7 8] = [19 22; 43 50]****BA = [5 6; 7 8] * [1 2; 3 4] = [23 34; 31 46]****B² = [5 6; 7 8] * [5 6; 7 8] = [61 70; 89 100]**

Adding these four matrices, we get:

**(A + B)² = [7 10; 15 22] + [19 22; 43 50] + [23 34; 31 46] + [61 70; 89 100] = [116 152; 220 292]**

As expected, both methods yield the same result.

### Applications

Understanding the square of a sum for matrices is crucial in various areas of mathematics and physics, including:

**Linear transformations:**Analyzing the behavior of linear transformations represented by matrices.**Eigenvalue problems:**Finding eigenvalues and eigenvectors of matrices.**Quadratic forms:**Representing quadratic functions in multi-dimensional spaces.**Differential equations:**Solving systems of differential equations.

By mastering this concept, you can gain a deeper understanding of matrix algebra and its numerous applications.