Understanding the Properties of Matrix Multiplication in Mathematics | Commutativity, Associativity, and Distributivity

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

Matrix multiplication is not commutative, meaning that the order in which you multiply two square matrices matters

Matrix multiplication is not commutative, meaning that the order in which you multiply two square matrices matters. In general, if you have two square matrices A and B, AB does not necessarily equal BA.

Matrix multiplication is associative, which means that if you have three square matrices A, B, and C, then (AB)C = A(BC). This property allows us to drop the parentheses when multiplying three or more matrices together.

Matrix multiplication is NOT distributive with respect to addition, which means that in general, (A + B)C does not equal AC + BC. However, it is distributive with respect to scalar multiplication, meaning that k(AB) = (kA)B = A(kB), where k is a scalar.

Here are some definitions related to matrix multiplication:
– Square matrix: A matrix in which the number of rows is equal to the number of columns.
– Matrix product: The result of multiplying two matrices together.
– Entry of a matrix: Each element in a matrix is called an entry. The entry in the i-th row and j-th column of a matrix is denoted by a(i, j).
– Row vector: A matrix with only one row is called a row vector.
– Column vector: A matrix with only one column is called a column vector.

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