Find the product:



An m×n matrix on the left can only be multiplied by an n×p matrix 
on the right and when they are multiplied the result is an m×p 
matrix: 



This is a 3×2 matrix times a 2×3 matrix, so m=3, n=2, p=3, so they
can be multiplied to give a 3×3 matrix.  Suppose the answer is this
3×3 matrix:



then:

 = 
 
A is in row 1, column 1, so multiply row 1 from 
the first matrix by column 1 from the second
matrix:

 =  =  = 

So replace the A by 6.

 = 
 
B is in row 1, column 2, so multiply row 1 from 
the first matrix by column 2 from the second
matrix:

 =  =  = 

So replace the B by -1.

 = 

C is in row 1, column 3, so multiply row 1 from 
the first matrix by column 3 from the second
matrix:

 =  =  = 

So replace the C by 5.

 = 

D is in row 2, column 1, so multiply row 2 from 
the first matrix by column 1 from the second
matrix:

 =  =  = 

So replace the D by 0.

 = 

E is in row 2, column 2, so multiply row 2 from 
the first matrix by column 2 from the second
matrix:

 =  =  = 

So replace the E by -8.

 = 

F is in row 2, column 3, so multiply row 2 from 
the first matrix by column 3 from the second
matrix:

 =  =  = 

So replace the F by 16.

 = 

G is in row 3, column 1, so multiply row 3 from 
the first matrix by column 1 from the second
matrix:

 =  =  = 

So replace the G by 4.

 = 

H is in row 3, column 2, so multiply row 3 from 
the first matrix by column 2 from the second
matrix:

 =  =  = 

So replace the H by -4.

 = 

I is in row 3, column 3, so multiply row 3 from 
the first matrix by column 3 from the second
matrix:

 =  =  = 

So replace the I by 10.

 = 

==============================================

Find the product: 


This is a 2×2 matrix times a 2×2 matrix, so m=2, n=2, p=2, so they
can be multiplied to give a 2×2 matrix.  Suppose the answer is this
2×2 matrix:



then:

=
 
A is in row 1, column 1, so multiply row 1 from 
the first matrix by column 1 from the second
matrix:

 =  = 

So replace the A by -35.

=

B is in row 1, column 2, so multiply row 1 from 
the first matrix by column 2 from the second
matrix:

 =  = =

So replace the B by 9.

=

C is in row 2, column 1, so multiply row 2 from 
the first matrix by column 1 from the second
matrix:

 =  = =

So replace the C by 10.

=

D is in row 2, column 2, so multiply row 2 from 
the first matrix by column 2 from the second
matrix:

 =  = =

So replace the D by -10.

=

Edwin