Question 122173
Solve the following system of equations by using the inverse of the coefficient matrix A. (AX=B) 

<pre>
  x + 4y = 22
-6x + 5y = 13

Form the matrix equation:

{{{(matrix(2,2,1,4,-6,5))(matrix(2,1,x,y))=(matrix(2,1,22,13))}}}
</pre>
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Now we must find the inverse of the coefficient matrix {{{(matrix(2,2,1,4,-6,5))}}}

To find the inverse of a 2x2 matrix: 

1. find the determinant of the matrix:
{{{abs(matrix(2,2,1,4,-6,5)))=(1)(5)-(4)(-6)}}}={{{5+24=29}}}
2. Swap the upper left and lower right elements
{{{(matrix(2,2,5,4,-6,1))}}}
3. Change the signs of the upper right and lower left elements
{{{(matrix(2,2,5,-4,6,1))}}}
4. Divide every term by the value of the determinant, which is 29.
{{{(matrix(2,2,5/29,-4/29,6/29,1/29))}}}
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<pre>
Now go back to the matrix equation

            {{{(matrix(2,2,1,4,-6,5))(matrix(2,1,x,y))=(matrix(2,1,22,13))}}}

Multiply the inverse matrix on the left of the
left side and also on the left of the right side;

{{{(matrix(2,2,5/29,-4/29,6/29,1/29))(matrix(2,2,1,4,-6,5))(matrix(2,1,x,y))=(matrix(2,2,5/29,-4/29,6/29,1/29))(matrix(2,1,22,13))}}}

I assume you know how to multiply matrices. If you
don't, post again asking how to. Multiply the first
two matrices on the left, and multiply the matrices on
the right:

            {{{(matrix(2,2,1,0,0,1))(matrix(2,1,x,y))=(matrix(2,1,2,5))}}}

We have the identity matrix on the left to multiply by
the matrix {{{(matrix(2,1,x,y))}}} which just gives:

                {{{(matrix(2,1,x,y))=(matrix(2,1,2,5))}}}

So we see that x = 2 and y = 5.

Edwin</pre>