Is matrix multiplication for square matrices commutative, associative, or distributive?
Matrix multiplication for square matrices is not commutative, but it is associative and distributes over addition
Matrix multiplication for square matrices is not commutative, but it is associative and distributes over addition.
Commutative property: Matrix multiplication is not commutative, which means that changing the order of multiplication will usually give different results. 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]
Simplifying, we have:
A * B = [19 22]
[43 50]
However, if we multiply B by A instead, we get:
B * A = [5*1 + 6*3 5*2 + 6*4]
[7*1 + 8*3 7*2 + 8*4]
Simplifying, we have:
B * A = [23 34]
[31 46]
As we can see, A * B is not equal to B * A, thus matrix multiplication is not commutative.
Associative property: Matrix multiplication is associative, which means that when you are multiplying three (or more) matrices together, the order of multiplication does not matter. For example, if we have matrices A, B, and C :
(A * B) * C = A * (B * C)
This property allows us to more easily perform calculations involving multiple matrix multiplications.
Distributive property: Matrix multiplication distributes over addition, just like how multiplication distributes over addition in basic arithmetic. For example, if we have matrices A, B, and C :
A * (B + C) = (A * B) + (A * C)
This property allows us to simplify calculations involving matrix multiplication.
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