Page 126 - Touhpad Ai

Multiplication of Matrices
The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.
Consider A = [a ], an m × n matrix and B = [b ] is an n x p matrix. Then the product of the matrices A and B is the matrix
ij
jk
th
th
C of order m × p. To get the (i, k)th element i.e c of the matrix C, we take the i row of A and k column of B, multiply
ik
them elementwise and take the sum of all these products. For example:
 5 1  26 0
A =  B =   
 23  2 2  13 4  2 3
×
×
equal  
AB =  
  2 3
×
Resultant Matrix, AB is
of the order 2 x 3
Take the first row of matrix A and first column of matrix B. Multiple the corresponding elements and then add them as
shown. Repeat this step for all columns of B.
51  26 0 
Step 1 A=   B =  
 2 3   1 3 4 
 11 ? ? 
⇒ (5 2 1 1) ⇒ 11 ⇒  
× +×
 ? ? ? 

Resultant Matrix, AB getting filled

51  26 0 
Step 2 A=   B =  
 23   13 4 

 11 33 ?
× +×
⇒ 5 6 1 3 ⇒ 33 ⇒  
 ? ? ? 
51  26 0 
Step 3 A =   B =  
 23   13 4 
 11 33 4
+×
×
⇒ 5 0 14 ⇒ 4 ⇒  
 ? ? ? 
Take the second row of matrix A and first column of matrix B. Multiple the corresponding elements and then add them
as shown. Repeat this step for all columns of B.
51  26 0 
Step 4 A =   B=  
 23   1 3 4 
 11 33 4
⇒ 2 2 + × ⇒ 7 ⇒  
×
3 1
 7 ? ? 
51  26 0 
Step 5 A =   B =  
 23   1 3 4 
 11 33 4
3 3 ⇒
×
⇒ 2 6 + × 21 ⇒  
 7 21 ? 
51  26 0 
Step 6 A =   B =  
 23   1 3 4 
 11 33 4   11 33 4 
×
3 4 ⇒
⇒ 2 0 + × 12 ⇒   Finally AB =  
 7 21 12   7 21 12  2 3
×
124 Touchpad Artificial Intelligence - XI