## The Square of a Matrix Sum: (A + B)² = A² + B² + 2AB

In linear algebra, understanding the behavior of matrices under various operations is crucial. One such operation is squaring a sum of matrices, represented by the equation:

**(A + B)² = A² + B² + 2AB**

This equation holds true **only** for certain types of matrices and under specific conditions. It is important to note that this is **not** a universal law for all matrices.

### Understanding the Equation

Let's break down the equation and the conditions for it to be valid:

**A and B:**These represent square matrices of the same order (i.e., they have the same number of rows and columns).**A² and B²:**These represent the square of each matrix. Squaring a matrix means multiplying it by itself.**AB:**This represents the matrix product of A and B.**2AB:**This represents the scalar multiplication of 2 with the product of matrices A and B.

### Conditions for the Equation to Hold True

The equation (A + B)² = A² + B² + 2AB is **not** universally true for all matrices. It holds true under the following conditions:

**Commutativity:**The matrices A and B must commute, meaning AB = BA. This condition is essential because matrix multiplication is generally not commutative.**Square Matrices:**Both A and B must be square matrices of the same order.

### Examples

**Example 1:** Let's consider two matrices:

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

**Verification:**

**AB = [[19, 22], [43, 50]]****BA = [[19, 22], [43, 50]]**

Since AB = BA, the matrices A and B commute. Therefore, the equation (A + B)² = A² + B² + 2AB holds true in this case.

**Example 2:** Let's consider two matrices:

**A = [[1, 2], [3, 4]]****B = [[5, 0], [0, 6]]**

**Verification:**

**AB = [[5, 12], [15, 24]]****BA = [[5, 0], [15, 24]]**

Since AB ≠ BA, the matrices A and B do not commute. Therefore, the equation (A + B)² = A² + B² + 2AB does **not** hold true in this case.

### Conclusion

The equation (A + B)² = A² + B² + 2AB holds true only when the matrices A and B commute and are both square matrices of the same order. It's crucial to remember this condition before applying the equation. Failing to do so can lead to incorrect results.