Question 244582
<pre><font size = 4 color = "indigo"><b>

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

First find the inverse of {{{(matrix(2,2,4,-5,1,2))}}}:

To do that:

1.  Find the value of its determinant, {{{abs(matrix(2,2,4,-5,1,2))=(4)(2)-(-5)(1) = 8+5 = 13}}}

2. Swap the upper left and lower right elements of {{{(matrix(2,2,4,-5,1,2))}}},
getting {{{(matrix(2,2,2,-5,1,4))}}}

3. Then change the signs of the upper right and lower left elements,
getting {{{(matrix(2,2,2,5,-1,4))}}}

4. Divide every element by the value of the determinant of the 
original matrix which we found to be 13 in step 1, getting
{{{(matrix(2,2,2/13,5/13,-1/13,4/13))}}}. This is the inverse of
the original matrix.

Left-multiply both sides of the given matrix equation:

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

by this inverse:

{{{
(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))}}} 

Use the associative principle to move the parentheses around
the first two 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))}}} 

Do the matrix multiplication:

{{{(matrix(2,2,(2/13)(4)+(5/13)(1), (2/13)(-5)+(5/13)(2),
(-1/13)(4)+(4/13)(1), (-1/13)(-5)+(4/13)(2) ))(matrix(2,1,m,n))=
(matrix(2,1,(2/13)(32)+(5/13)(-5), (-1/13)(32)+(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, 64/13-25/13, -32/13-20/13)) }}}

Simplify some more:

{{{ (matrix(2,2,13/13, 0, 0, 13/13 ))(matrix(2,1,m,n))=(matrix(2,1, 39/13, -52/13)) }}}

Simplify some more:

{{{ (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))}}}

Edwin</pre>