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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).

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    $\begingroup$ 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. $\endgroup$ Commented Jul 22, 2019 at 2:06
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    $\begingroup$ If you really want to know: en.wikipedia.org/wiki/Laplace_expansion#Proof $\endgroup$
    – EuxhenH
    Commented Jul 22, 2019 at 6:31
  • $\begingroup$ 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 $\endgroup$ Commented Jul 22, 2019 at 20:15

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