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### Inverse of a Matrix

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Date: 12/29/97 at 20:00:55
From: BMCcheer1

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

Beth
```

```
Date: 12/30/97 at 07:57:49
From: Doctor Jerry

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

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:

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/
```
Associated Topics:
High School Linear Algebra

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