Proof of the Laplace Expansion? I just learned about Laplace Expansion for determinant calculation in high-school, they taught me how to calculate minors cofactors and everything but they did not include the proof in the book. I have tried watching YouTube videos and there is a proof on Wikipedia which I cannot understand anything.
My teacher says she doesn't know the proof. Is the proof too hard for a high school student?
 A: First you need to understand odd and even permutation. Define $\sigma$ as the permutation of the first $n$ positive integers:
$$
\sigma=(\sigma_{1}, \sigma_{2},…,\sigma_{n})
$$
Define $s_{i}$ as the number of $j >i$ such that $\sigma_{i}>\sigma_{j}$. If $\sum_{i=1}^{n}{s_{i}}$ is odd we call $\sigma$ an odd permutation and even permutation otherwise. Then we define the $sgn$ function below:
$$
sgn(\sigma)=
\begin{cases}
\phantom{-}1\phantom{xx}\text{even}\phantom{x}\sigma\\
-1\phantom{xx}\text{odd}\phantom{x}\sigma
\end{cases}
$$
We also define a special permutations; $\tau^{j}$ as the permutation of the first $n$ positive integers except $j$. Notice the following:
$$
\begin{align}
&\text{if}\\
&\phantom{xx}\sigma^{*}=(j,\tau^{j}_{1},\tau^{j}_{2},…,\tau^{j}_{n-1})\\
\\
&\text{then}\\
&\phantom{xx}sgn(\sigma^{*})=\left(-1\right)^{j-1}sgn(\tau^{j})
\end{align}
$$
Now we move on to the definition of determinant using Leibniz formula:
$$
|A|=\sum_{all\phantom{x}\sigma}\left(sgn(\sigma)\prod_{i=1}^{n}{a_{i,\sigma_{i}}}\right)
$$
We can rewrite this equation as the following
$$
\begin{align}
|A|&=\sum_{i=1}^{n}{\left(a_{1,i}\sum_{all\phantom{x}\sigma^{*}}{\left(sgn(\sigma^{*})\prod_{j=1}^{n-1}{a_{(j+1),\tau^{i}_{j}}}\right)}\right)}\\
\\
&= \sum_{i=1}^{n}{\left((-1)^{i-1}\phantom{x}a_{1,i}\sum_{all\phantom{x}\tau^{i}}{\left(sgn(\tau^{i})\prod_{j=1}^{n-1}{a_{(j+1),\tau^{i}_{j}}}\right)}\right)}\\
\\
&= \sum_{i=1}^{n}{\left((-1)^{i-1}\phantom{x}a_{1,i}|A_{(1)(i)}\phantom{.}|\right)}
\end{align}
$$
Above, $A_{(1)(i)}$ is the matrice $A$ with the first row and $i$ -th column deleted. I think the last part is familiar with you since you have learned about minor, cofactor, etc..
A: YouTube - Proof of the Cofactor Expansion Theorem – Brian V 
This YouTube video might help you.
