SOLUTION: Solve a system using the inverse of a 3x3 matrix. 6x-5y-z=31, -x+2y+z=-6, 3x+3y+2z=13

6x-5y-z=31, 
-x+2y+z=-6, 
3x+3y+2z=13

We convert the system of equations:



to the matrix equation:




I will assume that you know how to find the inverse of the 
coefficient matrix.  If you don't, ask me to help you in the 
thank-you note form below, and I'll get back to you
by email. [No charge ever! I do this for fun!] 

It is:

 

Now we left multiply both sides of the matrix equation by
the inverse of the coefficient matrix:



I will also assume that you know how to multiply
matrices.  If you don't, ask me to help you in the 
thank-you note form below.

Multiplying the matrices on the right and
multiplying the first two on the left, we get:



That's the identity matrix on the left, which is what you
always get when you multiply a matrix by its inverse.
Multiplying the identity matrix by the column matrix of 
variables on the left we get



So x = 5, y= 0, z = -1.

Even though I did the entire problem, working with the
entire matrix equation, you can see that you can use a 
shortcut of just doing the right side only, which is to 
multiply the inverse of the coefficient matrix by the
column matrix of constants.  That's because in every 
case the left side will be the same as above.  

Edwin