(a+b)^2 Matrix Formula

Jasonbradley

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Jun 16, 2024

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The (A + B)^2 Matrix Formula: A Detailed Explanation

The formula (A + B)² is a common algebraic expression that can be applied to matrices as well. However, unlike scalar multiplication, where (a + b)² simply expands to a² + 2ab + b², the matrix equivalent is not as straightforward.

Why (A + B)² ≠ A² + 2AB + B² for Matrices

The reason why the simple expansion doesn't work for matrices lies in the non-commutative nature of matrix multiplication. In general, AB ≠ BA for matrices A and B. This means the order in which we multiply matrices matters, and we cannot simply rearrange terms as we would with scalars.

The Correct Formula

The correct expansion for (A + B)² in matrix algebra is:

(A + B)² = A² + AB + BA + B²

Let's break down each term:

• A²: This represents the matrix A multiplied by itself.
• AB: This represents the matrix multiplication of A and B.
• BA: This represents the matrix multiplication of B and A.
• B²: This represents the matrix B multiplied by itself.

Example

Let's illustrate with a simple example:

Suppose we have two 2x2 matrices:

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

To calculate (A + B)², we follow these steps:

1. Calculate A²:

   A² = A * A = [[1, 2], [3, 4]] * [[1, 2], [3, 4]] = [[7, 10], [15, 22]]
   

2. Calculate AB:

   AB = [[1, 2], [3, 4]] * [[5, 6], [7, 8]] = [[19, 22], [43, 50]]
   

3. Calculate BA:

   BA = [[5, 6], [7, 8]] * [[1, 2], [3, 4]] = [[23, 34], [31, 46]] 
   

4. Calculate B²:

   B² = B * B = [[5, 6], [7, 8]] * [[5, 6], [7, 8]] = [[67, 78], [91, 106]]
   

5. Add all the terms:

   (A + B)² = A² + AB + BA + B² = [[7, 10], [15, 22]] + [[19, 22], [43, 50]] + [[23, 34], [31, 46]] + [[67, 78], [91, 106]] = [[116, 144], [180, 224]]
   

Conclusion

The (A + B)² matrix formula is a useful tool in linear algebra, especially when working with matrix equations and transformations. It's important to remember that matrix multiplication is not commutative, so the order of multiplication matters. By understanding this difference, we can correctly apply the formula and avoid common pitfalls in matrix calculations.