Question

Find the inverse of A = [−5 2 2; −3 1 1; −9 3 4]

Answer 1

There are several ways to find the matrix, here is my method

1. find the determinant of A, I go det(A) = 1

2. flip the matrix along its top-left to bottom-right diagonal
to get
-5 -3 -9
2 1 3
2 1 4

3. replace each of the terms of this matrix by its cofactors
e.g -5 is replaced with 1x4-1x3 or 1
the 3 in the bottom row is replaced by -5x1 - (-3x2) = 1
you should have
1 2 0
-3 2 1
0 3 1

4. switch every second sign using the following patters
+ - +
- + -
+ - + , that is, when you see a + , leave the term alone, if you see a - for that position, change the sign of the term
to get

1 -2 0
3 -2 -1
0 -3 1

5. Finally divide each term of the matrix of step #4 by det(A), which we were lucky here , was 1

A^-1 =
1 -2 0
3 -2 -1
0 -3 1

here is a webpage which follows my method.
The order in which they show the steps differs from mine, but it does not matter.

http://www.mathsisfun.com/algebra/matrix-inverse-minors-cofactors-adjugate.html

Answer 2

My explanation of step 4 of my method is not as clear as I intended.
The webpage I gave you states it better and shows the pattern

Btw, if you had a 4 by 4 you better get several sheets of paper, since you have to bring it from a 4 by 4 to a 3 by 3, that alone would involve 16 calculations, and each of those would require 9 , so it becomes rather ridiculous.
For those I suggest some sort of computer applications. There are many of those.
here is one:
http://matrix.reshish.com/inverse.php

You might want to use it to check my answer.

Answer 3

Your cheating isn't helping you Hannah, I hope you know that this will be seen by the school, and there will be major punishment, and webmails going to your mother. I hope you don't mind that!