Question

Is matrix multiplication commutative?

Answer 1

Hint: Matrix is a rectangular array of numbers or expressions arranged in rows and columns. Matrix operations mainly involve three algebraic operations which are addition of matrices, subtraction of matrices, and multiplication of matrices and for multiplication of two matrices the number of columns of the first matrix must be equal to the number of rows of the second matrix.

Complete answer:
Addition, subtraction and multiplication are the basic operations on the matrix. To add or subtract matrices, these must be of identical order and for multiplication, the number of columns in the first matrix equals the number of rows in the second matrix.
For matrix multiplication to work, the columns of the second matrix have to have the same number of entries as do the rows of the first matrix. In particular, matrix multiplication is not commutative, you cannot switch the order of the factors and one of the biggest differences between real number multiplication and matrix multiplication is that matrix multiplication is not commutative.
In other words, in matrix multiplication, the order in which two matrices are multiplied matters.

Note: The key point is that to multiply two matrices, the number of columns of the first matrix must be equal to the number of rows of the second matrix, and after multiplication will have same number of rows as the first matrix and same number of columns of the second matrix. Hence, matrix multiplication is not commutative in general.