Is matrix multiplication for square matrices commutative, associative, or distributive?
Matrix multiplication for square matrices is not commutative, meaning that changing the order of the matrices will result in different products
Matrix multiplication for square matrices is not commutative, meaning that changing the order of the matrices will result in different products.
For example, let’s consider two square matrices A and B:
A = [1 2]
[3 4]
B = [5 6]
[7 8]
If we multiply A by B, we get:
A × B = [1×5 + 2×7 1×6 + 2×8]
[3×5 + 4×7 3×6 + 4×8]
= [19 22]
[43 50]
However, if we multiply B by A, we get:
B × A = [5×1 + 6×3 5×2 + 6×4]
[7×1 + 8×3 7×2 + 8×4]
= [27 34]
[31 40]
As you can see, the products A × B and B × A are not equal, indicating that matrix multiplication for square matrices is not commutative.
Matrix multiplication is associative, meaning that when we have three square matrices A, B, and C:
(A × B) × C = A × (B × C)
Distributivity, however, does not apply to matrix multiplication.
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