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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

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