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

Algebra ->  Matrices-and-determiminant -> SOLUTION: Find the product: {{{(matrix(3,2,3,-1,0,4,2,1))}}}{{{(matrix(2,3,2,-1,3,0,-2,4))}}} ` Find the product: {{{(matrix(2,2,3,-7,1,2))}}}{{{(matrix(2,2,0,-4,5,-3))}}}       Log On


   

Question 94381: Find the product:
%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29
`
Find the product:
%28matrix%282%2C2%2C3%2C-7%2C1%2C2%29%29%28matrix%282%2C2%2C0%2C-4%2C5%2C-3%29%29
Answer by Edwin McCravy(20081) About Me  (Show Source):
You can put this solution on YOUR website!
Find the product:

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29

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: 

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29

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:

%28matrix%283%2C3%2CA%2CB%2CC%2CD%2CE%2CF%2CG%2CH%2CI%29%29

then:

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29 = %28matrix%283%2C3%2CA%2CB%2CC%2CD%2CE%2CF%2CG%2CH%2CI%29%29
 
A is in row 1, column 1, so multiply row 1 from 
the first matrix by column 1 from the second
matrix:

%28matrix%281%2C2%2C3%2C-1%29%29%28matrix%282%2C1%2C2%2C0%29%29 = 3%2A2%2B-1%2A0 = 6-0 = 6

So replace the A by 6.

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29 = %28matrix%283%2C3%2C6%2CB%2CC%2CD%2CE%2CF%2CG%2CH%2CI%29%29
 
B is in row 1, column 2, so multiply row 1 from 
the first matrix by column 2 from the second
matrix:

%28matrix%281%2C2%2C3%2C-1%29%29%28matrix%282%2C1%2C-1%2C-2%29%29 = 3%2A-1%2B-1%2A-2 = -3%2B2 = -1

So replace the B by -1.

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29 = %28matrix%283%2C3%2C6%2C-1%2CC%2CD%2CE%2CF%2CG%2CH%2CI%29%29

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

%28matrix%281%2C2%2C3%2C-1%29%29%28matrix%282%2C1%2C3%2C4%29%29 = 3%2A3%2B-1%2A4 = 9-4 = 5

So replace the C by 5.

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29 = %28matrix%283%2C3%2C6%2C-1%2C5%2CD%2CE%2CF%2CG%2CH%2CI%29%29

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

%28matrix%281%2C2%2C0%2C4%29%29%28matrix%282%2C1%2C2%2C0%29%29 = 0%2A2%2B4%2A0 = 0%2B0 = 0

So replace the D by 0.

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29 = %28matrix%283%2C3%2C6%2C-1%2C5%2C0%2CE%2CF%2CG%2CH%2CI%29%29

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

%28matrix%281%2C2%2C0%2C4%29%29%28matrix%282%2C1%2C-1%2C-2%29%29 = 0%2A-1%2B4%2A-2 = 0-8 = -8

So replace the E by -8.

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29 = %28matrix%283%2C3%2C6%2C-1%2C5%2C0%2C-8%2CF%2CG%2CH%2CI%29%29

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

%28matrix%281%2C2%2C0%2C4%29%29%28matrix%282%2C1%2C3%2C4%29%29 = 0%2A3%2B4%2A4 = 0%2B16 = 16

So replace the F by 16.

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29 = %28matrix%283%2C3%2C6%2C-1%2C5%2C0%2C-8%2C16%2CG%2CH%2CI%29%29

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

%28matrix%281%2C2%2C2%2C1%29%29%28matrix%282%2C1%2C2%2C0%29%29 = 2%2A2%2B1%2A0 = 4%2B0 = 4

So replace the G by 4.

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29 = %28matrix%283%2C3%2C6%2C-1%2C5%2C0%2C-8%2C16%2C4%2CH%2CI%29%29

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

%28matrix%281%2C2%2C2%2C1%29%29%28matrix%282%2C1%2C-1%2C-2%29%29 = 2%2A-1%2B1%2A-2 = -2-2 = -4

So replace the H by -4.

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29 = %28matrix%283%2C3%2C6%2C-1%2C5%2C0%2C-8%2C16%2C4%2C-4%2CI%29%29

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

%28matrix%281%2C2%2C2%2C1%29%29%28matrix%282%2C1%2C3%2C4%29%29 = 2%2A3%2B1%2A4 = 6%2B4 = 10

So replace the I by 10.

%28matrix%283%2C2%2C3%2C-1%2C0%2C4%2C2%2C1%29%29%28matrix%282%2C3%2C2%2C-1%2C3%2C0%2C-2%2C4%29%29 = %28matrix%283%2C3%2C6%2C-1%2C5%2C0%2C-8%2C16%2C4%2C-4%2C10%29%29

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

Find the product: 
%28matrix%282%2C2%2C3%2C-7%2C1%2C2%29%29%28matrix%282%2C2%2C0%2C-4%2C5%2C-3%29%29

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:

%28matrix%282%2C2%2CA%2CB%2CC%2CD%29%29

then:

%28matrix%282%2C2%2C3%2C-7%2C1%2C2%29%29%28matrix%282%2C2%2C0%2C-4%2C5%2C-3%29%29=%28matrix%282%2C2%2CA%2CB%2CC%2CD%29%29
 
A is in row 1, column 1, so multiply row 1 from 
the first matrix by column 1 from the second
matrix:

%28matrix%281%2C2%2C3%2C-7%29%29%28matrix%282%2C1%2C0%2C5%29%29 = 3%2A0%2B-7%2A5 = -35

So replace the A by -35.

%28matrix%282%2C2%2C3%2C-7%2C1%2C2%29%29%28matrix%282%2C2%2C0%2C-4%2C5%2C-3%29%29=%28matrix%282%2C2%2C-35%2CB%2CC%2CD%29%29

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

%28matrix%281%2C2%2C3%2C-7%29%29%28matrix%282%2C1%2C-4%2C-3%29%29 = 3%2A-4%2B-7%2A-3 = -12%2B21=9

So replace the B by 9.

%28matrix%282%2C2%2C3%2C-7%2C1%2C2%29%29%28matrix%282%2C2%2C0%2C-4%2C5%2C-3%29%29=%28matrix%282%2C2%2C-35%2C9%2CC%2CD%29%29

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

%28matrix%281%2C2%2C1%2C2%29%29%28matrix%282%2C1%2C0%2C5%29%29 = 1%2A0%2B2%2A5 = 0%2B10=10

So replace the C by 10.

%28matrix%282%2C2%2C3%2C-7%2C1%2C2%29%29%28matrix%282%2C2%2C0%2C-4%2C5%2C-3%29%29=%28matrix%282%2C2%2C-35%2C9%2C10%2CD%29%29

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

%28matrix%281%2C2%2C1%2C2%29%29%28matrix%282%2C1%2C-4%2C-3%29%29 = 1%2A-4%2B2%2A-3 = -4-6=-10

So replace the D by -10.

%28matrix%282%2C2%2C3%2C-7%2C1%2C2%29%29%28matrix%282%2C2%2C0%2C-4%2C5%2C-3%29%29=%28matrix%282%2C2%2C-35%2C9%2C10%2C-10%29%29

Edwin