# Matrix multiplication order and its Properties

## what is Matrix Multiplication Order?

Matrix multiplication Order is a binary operation in which 2 matrices are multiply and produced a new matrix. The new matrix which is produced by 2 matrices is called the resultant matrix. Matrix multiplication is possible only if the number of columns in the first matrix is equal to the number of rows in the second matrix. The resulting matrix is also known as the product matrix. The resulting matrix has the number of rows of the first matrix and the number of columns of the second matrix. The product of two matrices A and B are denoted by AB.

### Short History about Matrix Multiplication Order

In 1812 the matrix multiplication order was first described by French mathematician Jacques Philippe Marie Binet to represent the composition of linear maps that are represented by matrices. Matrix multiplication is a basic tool of linear algebra.it is widely used in many fields. Matrix multiplication is applied to Mathematics, statistics, physics, economics, and engineering.

### Order of matrix multiplication

Multiplication of two matrices is possible when the number of columns in the first matrix is equal to the number of rows in the second matrix. the resultant matrix has the number of rows in the first matrix and the number of columns in the second matrix. as we know that the order of the matrix is defined as the number of rows multiplied by a number of columns. So the order of matrix multiplication order is that the number of rows of the first matrix and the number of columns of the second matrix.

### Example of matrix multiplication order

A=\begin{bmatrix}1&2&3\\1&2&3\end{bmatrix}, B=\begin{bmatrix}1&2&3&4\\2&1&2&3\\3&1&2&3\end{bmatrix}

Let us have two matrices A and B such that the order of matrix A is 2 –by 3 and the order of matrix B is 3 –by 4. the resultant matrix AB is produced and the order of that matrix is 2 –by- 4. Here the 2 is the number of rows in matrix A and 4 is the number of columns in matrix B.

### Properties of Matrix Multiplication Order

• Closure property
• Commutative property
• Distributive property
• Associative property
• Multiplicative property
• Identity property of addition
• Identity property of multiplication