# How to find C - the cofactor matrix of A?

1. Feb 26, 2012

### Cinitiator

1. The problem statement, all variables and given/known data
I need a way to find C - the cofactor matrix of A, assuming that A can be any arbitary matrix.

2. Relevant equations
See 1.

3. The attempt at a solution
Tried Googling without much success.

2. Feb 26, 2012

### genericusrnme

Do you know what the cofactor matrix actually is?

3. Feb 26, 2012

### HallsofIvy

Staff Emeritus
What do you mean by "find the cofactor matrix"? Are you looking for a formula so you can just plug in the values of a? Such a thing can be found but it would be terribly complicated. The simplest way to find it is to use the definition- each value, $C_{mn}$, is the mn-cofactor of A; that is, the determinant of the matrix you get by removing the mth row and nth column of A.

4. Feb 26, 2012

### Cinitiator

I was looking for a fomula to find a cofactor matrix of any other matrix. The reason why, is because I want to be able to inverse any matrices, without only using rules of thumb which work for simple 2x2 and 2x3 matrices.

As the inverse of a matrix A is 1/|A|*adj(A), and adj(A) is the transpose of a cofactor matrix C of A, I need it to find adjugates of any matrix, to later apply it to inverse matrices.

5. Feb 26, 2012

### rollcast

I'll try and explain it step by step, please correct me if I'm wrong.

Lets take a random matrix,

$$\begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix}$$

Then you take the cofactor of each element.

To find the cofactor take the element, eg. $A_{11}$ then you delete the row and column that the elment is in. Then you find the determinant of the resultant matrix.

So for our matrix,

$A_{11}$, so delete row 1 and column 1.

This leaves us with,

$$\begin{bmatrix} 4 & 5 \\ 0 & 6 \end{bmatrix}$$

Then take the determinant of that matrix.

$$\begin{vmatrix} 4 & 5 \\ 0 & 6 \end{vmatrix}$$ = ad-bc = 4 * 6 - 5 * 0 = 24.

So the first element, $A_{11}$, of the cofactor matrix is 24.

$$\begin{bmatrix}24 & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{bmatrix}$$

Just repeat the steps for the rest of the elements.

6. Feb 26, 2012

### Cinitiator

Thanks a lot, this helped.