The product of one matrix by another is not determined by the product of its respective elements.

Thus, the product of matrices A = (a_{ij})_{m x p} and B = (b_{ij})_{p x n} is the matrix C = (c_{ij})_{ m x n}where each element **ç _{ij}**is obtained by summing the products of the corresponding elements of the ith row of A by the elements of the ith column

**B**.

Let's multiply the matrices to understand how to get each element **ç _{ij}**:

1st row and 1st column

1st row and 2nd column

2nd row and 1st column

2nd row and 2nd column

Like this, .

Now look what would happen if you did the opposite, ie multiply B by A:

Therefore, .A, ie for matrix multiplication is not worth the commutative property.

Let's look at another example with the matrices :

From the definition, we have the product matrix A. B only exists if the number of columns of **THE** equals the number of lines of **B**:

The product matrix will have the number of rows of **Y **and the number of columns of **B (n)**:

If A

_{3 x 2}and B_{2 x 5 }then (A. B)_{3x5}If A

_{4x1}and B_{2x3}, then there is no productIf A

_{4 x 2}and B_{2x1}then (A. B)_{4x1}

## Properties

Once the conditions of existence for matrix multiplication are verified, the following properties apply:

a) associative: (A. B). C = A. (B. C)

b) distributive in relation to addition:. (B + C) = A. B + A. C or (A + B). C = A. C + B. Ç

c) neutral element:. I_{no }= I_{no} . A = A, where **I _{no}** the order identity matrix

**no**

We have seen that commutative property generally does not apply to matrix multiplication. Also not worth the cancellation of the product, ie: being 0 _{m x n} a null matrix, A .B = 0_{m x n} does not necessarily imply that A = 0_{m x n} or B = 0_{m x n}.

## Inverse matrix

Given an array **THE**square order **no**, if there is an array **THE'**of the same order as. A '= A'. A = I_{no} , then **THE'** is inverse matrix of **THE** . We represent the inverse matrix by **THE ^{-1 } .**