The (A  B)² Matrix Formula
The formula for the square of the difference of two matrices, (A  B)², is not as simple as squaring individual elements. This is because matrix multiplication follows specific rules. Here's a breakdown of how to calculate (A  B)²:
Understanding the Formula
The formula for (A  B)² is:
(A  B)² = (A  B)(A  B)
This means you need to multiply the matrix (A  B) by itself. However, matrix multiplication is not commutative, meaning AB ≠ BA. Therefore, we need to be careful about the order of multiplication.
StepbyStep Calculation

Calculate (A  B): Subtract the corresponding elements of matrices A and B. This results in a new matrix, let's call it C:
 C = A  B

Multiply (A  B) by itself: Now, multiply matrix C by itself:
 (A  B)² = C * C

Perform Matrix Multiplication: Apply the rules of matrix multiplication to calculate C * C.
Example
Let's consider two matrices:
 A = [[1, 2], [3, 4]]
 B = [[5, 6], [7, 8]]

Calculate (A  B):
 C = A  B = [[1  5, 2  6], [3  7, 4  8]] = [[4, 4], [4, 4]]

Multiply (A  B) by itself:
 (A  B)² = C * C = [[4, 4], [4, 4]] * [[4, 4], [4, 4]]

Perform Matrix Multiplication:
 (A  B)² = [[(4)(4) + (4)(4), (4)(4) + (4)(4)], [(4)(4) + (4)(4), (4)(4) + (4)(4)]] = [[32, 32], [32, 32]]
Therefore, (A  B)² = [[32, 32], [32, 32]].
Key Points
 Remember that matrix multiplication is not commutative.
 The order of multiplication matters when calculating (A  B)².
 The resulting matrix will have the same dimensions as matrices A and B.
This formula is crucial for various matrix operations and is frequently used in linear algebra, calculus, and other mathematical fields.