# What exactly is a cofactor and how is the sign chart derived?

Given

$$\begin{equation*} \begin{bmatrix} a_{11} & \dots & a_{1j} \\ \vdots & \ddots & \vdots \\ a_{i1} & \dots & a_{ij} \end{bmatrix} \end{equation*}$$

I know the cofactor is $$A_{ij}=M_{ij}(-1)^{i+j}$$ where $$M_{ij}$$ is the minor of an element. But what does that even mean? What does it mean to take the cofactor of a matrix, is there a geometric visualization of this or anything?

Second, where does the sign chart below come from, and why do I have to use it to take the row cofactor sum to find the determinant of the matrix? $$\begin{equation*} \begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix} \end{equation*}$$

I have a decent understanding of why $$\Delta x=\Delta_1$$, but it's cofactor expansions and the sign chart that I don't understand (but can do/use).

• Think of the cofactor of an entry as the derivative of the determinant with respect to that entry; it tells you (up to sign) how fast the determinant changes when you changes that entry. Commented Jul 22, 2019 at 2:06
• If you really want to know: en.wikipedia.org/wiki/Laplace_expansion#Proof Commented Jul 22, 2019 at 6:31
• I think of the sign chart as related to the property of determinants that switching adjacent rows (or columns) in a matrix switches the sign of the determinant Commented Jul 22, 2019 at 20:15