How to find C - the cofactor matrix of A?

[Cinitiator]

Homework Statement

I need a way to find C - the cofactor matrix of A, assuming that A can be any arbitary matrix.

Homework Equations

See 1.

The Attempt at a Solution

Tried Googling without much success.

 

[genericusrnme]

Do you know what the cofactor matrix actually is?

 

[HallsofIvy]

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.

 

[Cinitiator]

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.

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.

 

[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.

 

[Cinitiator]

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.

Thanks a lot, this helped.