Question 710737
<pre><font size = 4 color = "indigo"><b>
This is really the matrix abbreviation for the system of equtions:  

{{{system(4m-5n=32,m+2n=-5)}}}

{{{(matrix(2,2,4,-5,1,2))(matrix(2,1,m,n))}}}{{{""=""}}}{{{(matrix(2,1,32,-5))}}}

Now we need to find the inverse of the coefficient matrix

{{{(matrix(2,2,4,-5,1,2))}}}

To find the inverse of a 2x2 matrix:

1. Interchange the upper left and lower right elements:

{{{(matrix(2,2,2,-5,1,4))}}}

2. Multiply the upper right and lower left elements by -1:

{{{(matrix(2,2,2,5,-1,4))}}}

3. Find the determinant of this matrix:

{{{abs(matrix(2,2,2,5,-1,4))}}}{{{""=""}}}{{{(2)(4)-(5)(-1)}}}{{{""=""}}}{{{8+5}}}{{{""=""}}}{{{13}}}

4. Divide every element of {{{(matrix(2,2,2,5,-1,4))}}} by this value:

{{{(matrix(2,2,2/13,5/13,-1/13,4/13))}}}

That is the inverse of the coefficient matrix.

Left-multiply both sides of the matrix
equation:

{{{(matrix(2,2,4,-5,1,2))(matrix(2,1,m,n))}}}{{{""=""}}}{{{(matrix(2,1,32,-5))}}}

by the inverse of the coefficient matrix:

{{{

(matrix(2,2,2/13,5/13,-1/13,4/13))

( (matrix(2,2,4,-5,1,2))(matrix(2,1,m,n)) )}}}{{{""=""}}}{{{


(matrix(2,2,2/13,5/13,-1/13,4/13))
(matrix(2,1,32,-5))}}}

Since matrix multiplication is associative, we can move
the parentheses from around the 2nd and 3rd matrices
on the left to around the 1st and 2nd matrices on the
left:

{{{

((matrix(2,2,2/13,5/13,-1/13,4/13))

(matrix(2,2,4,-5,1,2)))(matrix(2,1,m,n))}}}{{{""=""}}}{{{

(matrix(2,2,2/13,5/13,-1/13,4/13))
(matrix(2,1,32,-5))}}}

Now we multiply the two matrices on the far
left and the far right:  

{{{

( matrix(2,2,

expr(2/13)(4)+expr(5/13)(1),expr(2/13)(-5)+expr(5/13)(2),expr(-1/13)(4)+expr(4/13)(1),expr(-1/13)(-5)+expr(4/13)(2)))

(matrix(2,1,m,n))}}}{{{""=""}}}{{{(matrix(2,1,expr(2/13)(32)+expr(5/13)(-5),expr(-1/13)(32)+expr(4/13)(-5)))}}}

Simplify:

{{{

( matrix(2,2,

8/13+5/13,-10/13+10/13,4/13-4/13,5/13+8/13))

(matrix(2,1,m,n))}}}{{{""=""}}}{{{(matrix(2,1,64/3-25/13,-32/13-20/13))

}}}

{{{

( matrix(2,2,

13/13,0/13,0/13,13/13))

(matrix(2,1,m,n))}}}{{{""=""}}}{{{(matrix(2,1,39/13,-52/13))

}}}

{{{( matrix(2,2,

1,0,0,1))

(matrix(2,1,m,n))}}}{{{""=""}}}{{{(matrix(2,1,3,-4))

}}}

Multiply the matrices on the left:

{{{(matrix(2,1,1*m+0*n, 0*m+1*n))}}}{{{""=""}}}{{{(matrix(2,1,3,-4))}}} 

Simplify:

{{{(matrix(2,1,m,n))}}}{{{""=""}}}{{{(matrix(2,1,3,-4))}}} 

So the solution is 

{{{m=3}}}, {{{n=-4}}}

Edwin</pre>