The Associative Property Of Matrix Multiplication For Square Matrices: Non-Commutative And Non-Distributive Properties Explained.

Is matrix multiplication for square matrices commutative, associative, or distributive?

only associative and distributive

Matrix multiplication for square matrices is associative, but not commutative or distributive.

Associative means that the way the matrices are grouped does not change the result. For example, if we have three square matrices A, B, and C of the same size, then:

(A*B)*C = A*(B*C)

This property allows us to efficiently perform calculations with long chains of matrix multiplications without having to worry about the order of multiplication.

However, matrix multiplication is not commutative. This means that the order of multiplication matters. For example, if we have two square matrices A and B of the same size, then in general:

A*B ≠ B*A

In other words, multiplying A by B will usually result in a different matrix than multiplying B by A.

Matrix multiplication is also not distributive. This means that we can’t distribute a matrix across the sum or difference of two other matrices in the same way that we can with real numbers. For example:

A*(B+C) ≠ A*B + A*C

In general, the product of two matrices A and B is defined as a new matrix C, where the i,j entry of C is the sum of the products of the ith row of A and the jth column of B. This operation satisfies the associative property, but not the commutative or distributive properties.

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