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# Matrix multiplication

Definition.
The result of the multiplication of matrices Am×n and Bn×k the matrix Cm×k such that the element of the matrix C, standing in the i-th row and j-th column (cij), is equal to the sum of products of elements of the i-th row of the matrix A by the corresponding elements j-th column of matrix B:

cij = ai1 · b1j + ai2 · b2j + ... + ain · bnj

Note.
Two matrices can be multiplied with each other when the number of columns of the first matrix equals the number of rows of the second matrix.

## Properties of matrix multiplication

• (A · B) · C= A · (B · C) - matrix multiplication is associative;
• (z · A) · B= z · (A · B), where z - number;
• A · (B + C) = A · B + A · C - matrix multiplication is distributive;
• In · Anm = Anm · Im= Anm - matrix multiplication by identity matrix;
• A · B ≠ B · A - in general, the matrix multiplication is not commutative.
• The product of two matrices is a matrix with as many rows as they have left the matrix, and as many columns as they have the right matrix.

## Examples of matrix multiplication

Example 1.
 Find the matrix C is the product of matrices A = 4 2 and B = 3 1 . 9 0 -3 4

Solution:

 С = A · B = 4 2 · 3 1 = 6 12 9 0 -3 4 27 9

The elements of the matrix C are calculated as follows:

c11 = a11·b11 + a12·b21 = 4·3 + 2·(-3) = 12 - 6 = 6

c12 = a11·b12 + a12·b22 = 4·1 + 2·4 = 4 + 8 = 12

c21 = a21·b11 + a22·b21 = 9·3 + 0·(-3) = 27 + 0 = 27

c22 = a21·b12 + a22·b22 = 9·1 + 0·4 = 9 + 0 = 9

Example 2
Find the matrix C is the product of matrices A = 2 1 -3 0 4 -1
and B = 5 -1 6 -3 0 7
.

Solution:

C = A · B = 2 1 -3 0 4 -1
· 5 -1 6 -3 0 7
= 7 -2 19 -15 3 -18 23 -4 17

The elements of the matrix C are calculated as follows:

c11 = a11·b11 + a12·b21 = 2·5 + 1·(-3) = 10 - 3 = 7

c12 = a11·b12 + a12·b22 = 2·(-1) + 1·0 = -2 + 0 = -2

c13 = a11·b13 + a12·b23 = 2·6 + 1·7 = 12 + 7 = 19

c21 = a21·b11 + a22·b21 = (-3)·5 + 0·(-3) = -15 + 0 = -15

c22 = a21·b12 + a22·b22 = (-3)·(-1) + 0·0 = 3 + 0 = 3

c23 = a21·b13 + a22·b23 = (-3)·6 + 0·7 = -18 + 0 = -18

c31 = a31·b11 + a32·b21 = 4·5 + (-1)·(-3) = 20 + 3 = 23

c32 = a31·b12 + a22·b22 = (4)·(-1) + (-1)·0 = -4 + 0 = -4

c33 = a31·b13 + a32·b23 = 4·6 + (-1)·7 = 24 - 7 = 17