%5E2-matrix-formula.jpeg "(a-b)^2 Matrix Formula")
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.
Step-by-Step Calculation
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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
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Multiply (A - B) by itself: Now, multiply matrix C by itself:
- (A - B)² = C * C
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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]]
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Calculate (A - B):
- C = A - B = [[1 - 5, 2 - 6], [3 - 7, 4 - 8]] = [[-4, -4], [-4, -4]]
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Multiply (A - B) by itself:
- (A - B)² = C * C = [[-4, -4], [-4, -4]] * [[-4, -4], [-4, -4]]
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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.