The Misconception of (a + b)² = a² + b² in Matrix Multiplication
The common algebraic identity (a + b)² = a² + 2ab + b² doesn't hold true when working with matrix multiplication. This is a crucial distinction to remember as it leads to common errors in linear algebra.
Why the Formula Doesn't Apply:
Matrix multiplication is not commutative, meaning the order of multiplication matters. This is different from standard scalar multiplication where ab = ba. As a result, the distributive property of multiplication works differently for matrices.
Let's break it down:
 (A + B)² is equivalent to (A + B)(A + B).
 Expanding this using the distributive property, we get:
 A(A + B) + B(A + B)
 AA + AB + BA + BB
 A² + AB + BA + B²
Notice that AB and BA are not necessarily equal. Therefore, the terms AB and BA cannot be combined to form a single term like 2AB.
Consequences of the Misconception:
Using the incorrect formula (a + b)² = a² + b² in matrix multiplication can lead to significant errors in calculations and analysis. It's essential to understand the correct expansion and apply the distributive property accurately.
Example:
Consider two matrices:

A = [1 2] [3 4]

B = [5 6] [7 8]
Let's calculate (A + B)²:

A + B = [6 8] [10 12]

(A + B)² = (A + B)(A + B) = [6 8] [6 8] = [96 112] [10 12] [10 12] [112 136]
Now, let's see what happens if we use the incorrect formula:

A² = [1 2] [1 2] = [5 8] [3 4] [3 4] [17 26]

B² = [5 6] [5 6] = [61 78] [7 8] [7 8] [78 100]

A² + B² = [5 8] + [61 78] = [66 86] [17 26] [78 100] [95 126]
As you can see, the result obtained using the incorrect formula (A² + B²) is significantly different from the correct result of (A + B)².
Key Takeaways:
 The identity (a + b)² = a² + 2ab + b² does not apply to matrix multiplication.
 Always remember that AB ≠ BA in matrix multiplication.
 Use the distributive property correctly when expanding expressions involving matrix addition and multiplication.
 Avoid using the incorrect formula to prevent errors in your linear algebra calculations.