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

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

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.

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!

To find the inverse of a matrix A, we need to follow these steps:

Step 1: Calculate the determinant of matrix A. Let's denote it as det(A).

Step 2: If the determinant (det(A)) is equal to zero, then the inverse does not exist. In this case, we cannot continue with finding the inverse.

Step 3: If the determinant is not zero, we can proceed to find the adjugate matrix of A. The adjugate matrix of A, denoted as adj(A), is obtained by taking the transpose of the cofactor matrix of A.

Step 4: Finally, to calculate the inverse of A, we divide the adjugate matrix adj(A) by the determinant det(A). So the inverse of A is given by A^(-1) = (1/det(A)) * adj(A).

Now let's find the inverse of the matrix A = [-5 2 2; -3 1 1; -9 3 4]:

Step 1: Calculate the determinant of A.

det(A) = (-5 * (1 * 4 - 1 * 3)) - (2 * (-3 * 4 - 1 * (-9))) + (2 * (-3 * 3 - (-9) * 2))
= (-5 * 1) - (2 * (-12 + 9)) + (2 * (-9 - (-18)))
= -5 + 6 + (-6)
= -5

Step 2: Since the determinant of A is not zero, we can continue with finding the inverse.

Step 3: Calculate the cofactor matrix of A. The cofactor of each element is calculated by applying a sign pattern and finding the determinant of the 2x2 submatrix obtained by removing the row and column that contain the respective element. The cofactor matrix of A is then obtained by putting the cofactors in their respective positions.

The cofactor matrix of A = [-(1 * 4 - 1 * 3) (1 * 4 - 1 * (-9)) -(1 * (-9) - (-3) * (-3));
(2 * 4 - 2 * 3) -(2 * 4 - 2 * (-9)) (2 * (-9) - (-3) * (-3));
-(2 * 3 - (-9) * 1) (2 * 3 - (-9) * (-3)) -(2 * (-9) - (-3) * 1)]
= [-1 13 -6;
2 -26 15;
-3 -21 19]

The adjugate matrix of A is obtained by taking the transpose of the cofactor matrix:

adj(A) = [-1 2 -3;
13 -26 -21;
-6 15 19]

Step 4: Calculate the inverse of A by dividing adj(A) by det(A).

A^(-1) = (1/det(A)) * adj(A)
= (1/-5) * [-1 2 -3;
13 -26 -21;
-6 15 19]
= [1/5 -2/5 3/5;
-13/5 26/5 21/5;
6/5 -15/5 -19/5]

Therefore, the inverse of the matrix A = [-5 2 2;
-3 1 1;
-9 3 4] is:

A^(-1) = [1/5 -2/5 3/5;
-13/5 26/5 21/5;
6/5 -15/5 -19/5]