SOLUTION: Solve the matrix equation: {{{(matrix(2,2,4,-5,1,2))(matrix(2,1,m,n))=(matrix(2,1,32,-5))}}} Please Help

Algebra ->  Matrices-and-determiminant -> SOLUTION: Solve the matrix equation: {{{(matrix(2,2,4,-5,1,2))(matrix(2,1,m,n))=(matrix(2,1,32,-5))}}} Please Help       Log On


   

Question 244582: Solve the matrix equation:

Please Help
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!


 

First find the inverse of %28matrix%282%2C2%2C4%2C-5%2C1%2C2%29%29:

To do that:

1.  Find the value of its determinant, abs%28matrix%282%2C2%2C4%2C-5%2C1%2C2%29%29=%284%29%282%29-%28-5%29%281%29+=+8%2B5+=+13

2. Swap the upper left and lower right elements of %28matrix%282%2C2%2C4%2C-5%2C1%2C2%29%29,
getting %28matrix%282%2C2%2C2%2C-5%2C1%2C4%29%29

3. Then change the signs of the upper right and lower left elements,
getting %28matrix%282%2C2%2C2%2C5%2C-1%2C4%29%29

4. Divide every element by the value of the determinant of the 
original matrix which we found to be 13 in step 1, getting
%28matrix%282%2C2%2C2%2F13%2C5%2F13%2C-1%2F13%2C4%2F13%29%29. This is the inverse of
the original matrix.

Left-multiply both sides of the given matrix equation:



by this inverse:

 

Use the associative principle to move the parentheses around
the first two matrices on the left:

 

Do the matrix multiplication:



Simplify:



Simplify some more:



Simplify some more:



Multiply the matrices on the left:

%28matrix%282%2C1%2C1%2Am%2B0%2An%2C0%2Am%2B1%2An%29%29=%28matrix%282%2C1%2C3%2C-4%29%29

Simplify:

%28matrix%282%2C1%2Cm%2Cn%29%29=%28matrix%282%2C1%2C3%2C-4%29%29

Edwin