Introduction
Matrix multiplication is an important operation in linear algebra and is widely used in business mathematics, economics, computer science, and physics. One interesting and important fact about matrix multiplication is that it is not commutative. This means that if you have two matrices A and B, then A × B ≠ B × A in most cases. In this answer, we will explain in simple language why matrix multiplication is not commutative.
What is Commutativity?
An operation is called commutative if changing the order of the operands does not change the result.
For example:
- In addition: 2 + 3 = 3 + 2 ✅
- In multiplication: 4 × 5 = 5 × 4 ✅
- But in matrix multiplication: A × B is usually not equal to B × A ❌
Understanding Matrix Multiplication
Let’s take a basic example of two matrices:
A = [ [1, 2], [3, 4] ]
B = [ [5, 6], [7, 8] ]
Multiply A × B
Step-by-step:
A × B = [
(1×5 + 2×7), (1×6 + 2×8)
(3×5 + 4×7), (3×6 + 4×8)
] = [
(5+14), (6+16)
(15+28), (18+32)
] = [
19, 22
43, 50
]
Now Multiply B × A
B × A = [
(5×1 + 6×3), (5×2 + 6×4)
(7×1 + 8×3), (7×2 + 8×4)
] = [
(5+18), (10+24)
(7+24), (14+32)
] = [
23, 34
31, 46
]
A × B ≠ B × A — Clearly, the results are different.
Why Is Matrix Multiplication Not Commutative?
There are several reasons:
1. Matrix Dimensions
To multiply two matrices A and B:
- The number of columns in A must equal the number of rows in B.
So, even if A × B is possible, B × A may not be possible.
Example: If A is 2×3 and B is 3×4, A × B is valid (resulting in a 2×4 matrix), but B × A is not valid because 4 ≠ 2.
2. Even if Sizes Match, Order Matters
Even when both A × B and B × A are possible, they may give different results because matrix multiplication involves row-by-column combinations, and changing the order changes the rows and columns involved.
3. Different Interpretations in Applications
In practical uses like economics, physics, and computer graphics, the meaning of the matrices changes with order. For example:
- One matrix might represent data
- Another might represent a transformation (like scaling, rotating, etc.)
Changing the order changes the meaning and therefore the result.
Exceptions: When is Matrix Multiplication Commutative?
There are rare cases when A × B = B × A:
- If A and B are identity matrices (e.g., I × A = A × I)
- If A = B and both are diagonal matrices with same dimensions
- When A and B commute due to special structure (e.g., scalar matrices)
Conclusion
Matrix multiplication is generally not commutative because:
- Dimensions may not allow reversal
- Even if allowed, the numerical result will usually be different
- The process of combining rows and columns depends on the order
This property is important in real-world applications. For example, in business mathematics, if matrices represent operations like cost or production, changing the order of application may completely change the results. Understanding this helps prevent errors in data analysis, financial modeling, and engineering calculations.