Question 443287
I am stuck on a project, and am desperate for help. My teacher denied me any help during class, so I'm kind of packed into a corner.
1. I have to write an augmented matrix using this system : 2x=5y 3y+x=-11
2.After I find the augmented matrix, I have to find two matrices that multiply to a product of the augmented matrix.
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{{{system(2x=5y,3y+x=-11)}}}

Get the equations in general form:

{{{system(2x-5y=0,1x+3y=-11)}}}

The augmented matrix is just an array of all the numbers
in the system without the letters or equal signs:


{{{(matrix(2,3,

 2,-5,0,
 1, 3,-11))}}}

That's a 2x3 matrix.

To find two matrices that have a 2x3 matrix as their
product, the first could be a 2x2 and the second a 2x3
matrix. 

Let's make up an arbitrary 2x2 matrix, say

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

Then we'll let the 2x3 matrix be:

{{{(matrix(2,3,p,q,r,s,t,u))}}}

So we have

{{{(matrix(2,2,2,3,1,-4))(matrix(2,3,p,q,r,s,t,u))

= (matrix(2,3,2p+3s,2q+3t,2r+3u,p-4s,q-4t,r-4u)) = (matrix(2,3,2,-5,0,1,3,-11))}}}

Equating elements of the two 2x3 matrices:, we have 
these 3 systems of equations:

{{{system(2p+3s=2,p-4s=1)}}}, {{{system(2q+3t=-5,q-4t=3)}}}, {{{system(2r+3u=0,r-4u=-11)}}}, 

We solve those 3 systems and get p=1,s=0, q=-1,t=-1,r=-3,u=2

So one possible answer is

{{{(matrix(2,2,2,3,1,-4))(matrix(2,3,p,q,r,s,t,u))
= (matrix(2,3,2,-5,0,1,3,-11))}}}

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

= (matrix(2,3,2,-5,0,1,3,-11))}}}
  
But you could make up any (non-singular) 2x2 matrix
and find a 2x3 matrix so that the product would be
equal to the augmented matrix.  There are many possible
answers, in fact infinitely many.

Edwin</pre>