Question 94381
<pre>
Find the product:

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}}

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: 

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}}

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:

{{{(matrix(3,3,A,B,C,D,E,F,G,H,I))}}}

then:

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}} = {{{(matrix(3,3,A,B,C,D,E,F,G,H,I))}}}
 
A is in row 1, column 1, so multiply row 1 from 
the first matrix by column 1 from the second
matrix:

{{{(matrix(1,2,3,-1))}}}{{{(matrix(2,1,2,0))}}} = {{{3*2+-1*0}}} = {{{6-0}}} = {{{6}}}

So replace the A by 6.

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}} = {{{(matrix(3,3,6,B,C,D,E,F,G,H,I))}}}
 
B is in row 1, column 2, so multiply row 1 from 
the first matrix by column 2 from the second
matrix:

{{{(matrix(1,2,3,-1))}}}{{{(matrix(2,1,-1,-2))}}} = {{{3*-1+-1*-2}}} = {{{-3+2}}} = {{{-1}}}

So replace the B by -1.

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}} = {{{(matrix(3,3,6,-1,C,D,E,F,G,H,I))}}}

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

{{{(matrix(1,2,3,-1))}}}{{{(matrix(2,1,3,4))}}} = {{{3*3+-1*4}}} = {{{9-4}}} = {{{5}}}

So replace the C by 5.

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}} = {{{(matrix(3,3,6,-1,5,D,E,F,G,H,I))}}}

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

{{{(matrix(1,2,0,4))}}}{{{(matrix(2,1,2,0))}}} = {{{0*2+4*0}}} = {{{0+0}}} = {{{0}}}

So replace the D by 0.

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}} = {{{(matrix(3,3,6,-1,5,0,E,F,G,H,I))}}}

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

{{{(matrix(1,2,0,4))}}}{{{(matrix(2,1,-1,-2))}}} = {{{0*-1+4*-2}}} = {{{0-8}}} = {{{-8}}}

So replace the E by -8.

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}} = {{{(matrix(3,3,6,-1,5,0,-8,F,G,H,I))}}}

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

{{{(matrix(1,2,0,4))}}}{{{(matrix(2,1,3,4))}}} = {{{0*3+4*4}}} = {{{0+16}}} = {{{16}}}

So replace the F by 16.

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}} = {{{(matrix(3,3,6,-1,5,0,-8,16,G,H,I))}}}

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

{{{(matrix(1,2,2,1))}}}{{{(matrix(2,1,2,0))}}} = {{{2*2+1*0}}} = {{{4+0}}} = {{{4}}}

So replace the G by 4.

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}} = {{{(matrix(3,3,6,-1,5,0,-8,16,4,H,I))}}}

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

{{{(matrix(1,2,2,1))}}}{{{(matrix(2,1,-1,-2))}}} = {{{2*-1+1*-2}}} = {{{-2-2}}} = {{{-4}}}

So replace the H by -4.

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}} = {{{(matrix(3,3,6,-1,5,0,-8,16,4,-4,I))}}}

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

{{{(matrix(1,2,2,1))}}}{{{(matrix(2,1,3,4))}}} = {{{2*3+1*4}}} = {{{6+4}}} = {{{10}}}

So replace the I by 10.

{{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}} = {{{(matrix(3,3,6,-1,5,0,-8,16,4,-4,10))}}}

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

Find the product: 
{{{(matrix(2,2,3,-7,1,2))}}}{{{(matrix(2,2,0,-4,5,-3))}}}

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:

{{{(matrix(2,2,A,B,C,D))}}}

then:

{{{(matrix(2,2,3,-7,1,2))}}}{{{(matrix(2,2,0,-4,5,-3))}}}={{{(matrix(2,2,A,B,C,D))}}}
 
A is in row 1, column 1, so multiply row 1 from 
the first matrix by column 1 from the second
matrix:

{{{(matrix(1,2,3,-7))}}}{{{(matrix(2,1,0,5))}}} = {{{3*0+-7*5}}} = {{{-35}}}

So replace the A by -35.

{{{(matrix(2,2,3,-7,1,2))}}}{{{(matrix(2,2,0,-4,5,-3))}}}={{{(matrix(2,2,-35,B,C,D))}}}

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

{{{(matrix(1,2,3,-7))}}}{{{(matrix(2,1,-4,-3))}}} = {{{3*-4+-7*-3}}} = {{{-12+21}}}={{{9}}}

So replace the B by 9.

{{{(matrix(2,2,3,-7,1,2))}}}{{{(matrix(2,2,0,-4,5,-3))}}}={{{(matrix(2,2,-35,9,C,D))}}}

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

{{{(matrix(1,2,1,2))}}}{{{(matrix(2,1,0,5))}}} = {{{1*0+2*5}}} = {{{0+10}}}={{{10}}}

So replace the C by 10.

{{{(matrix(2,2,3,-7,1,2))}}}{{{(matrix(2,2,0,-4,5,-3))}}}={{{(matrix(2,2,-35,9,10,D))}}}

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

{{{(matrix(1,2,1,2))}}}{{{(matrix(2,1,-4,-3))}}} = {{{1*-4+2*-3}}} = {{{-4-6}}}={{{-10}}}

So replace the D by -10.

{{{(matrix(2,2,3,-7,1,2))}}}{{{(matrix(2,2,0,-4,5,-3))}}}={{{(matrix(2,2,-35,9,10,-10))}}}

Edwin</pre>