Inverse of a Matrix

Date: 12/29/97 at 20:00:55
From: BMCcheer1
Subject: *please help*

Hello, my name is Beth Catanzaro, and I am a 9th grade honors algebra 
student. I attend Coventry Middle School in Coventry Rhode Island and 
during vacation, am currently working on an important, but extremely 
difficult algebra project. This is where I run into a few problems and 
hope you can help. 

First off, the teacher has not taught us the material we are working 
with. He felt it would benefit us more to give out a packet containing 
all (or most) of the information we would need to answer his written 
questions. This is where my problems start. My first question as of 
now would be, 

"How do you figure out the inverse of a 2 x 2 matrix, such as  
[5  4]
[2  2] ?

The question asks to find the determinant, which I already know is 
ad-bc, but the explanation stops after that, so I don't know what to 
do next. If you could please help me, I would greatly appreciate it.  

Beth

Date: 12/30/97 at 07:57:49
From: Doctor Jerry
Subject: Re: *please help*

Hi Beth,

I'll assume that you know that in a system of equations you can do 
three things without changing the set of solutions.

 1. multiply any equation by a nonzero constant
 2. interchange any two equations
 3. add to any equation a multiple of another equation

From what you've said, I'll also assume that for the system

  a*x+b*y = r
  c*x+d*y = s

you know about augmented matrix

  a  b  r

  c  d  s

and that corresponding to 1, 2, and 3 above, there are the three row 
operations

   i. multiply any row by a nonzero constant
  ii. interchange any two rows
 iii. add to any row a multiple of another row

Here's a method for calculating the inverse of the coeffcient matrix:

Start with the special augmented matrix

  a  b  1  0

  c  d  0  1

This is the coefficient matrix to which has been adjoined the identity 
matrix. Now, do row operations on this matrix, working until you 
obtain

  1  0  A  B

  0  1  C  D

The matrix on the right is the inverse of the original coefficient 
matrix.

This method is a very efficient method of calculating the inverse.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/