\section{Summary -- Important Facts about Determinants}

\begin{definition}
	\hfill\\
	The \textbf{determinant} of an $n \times n$ matrix $A$ having entries from a field $\F$ is a scalar in $\F$, denoted by $\det(A)$ or $|A|$, and can be computed in the following manner:

	\begin{enumerate}
		\item If $A$ is $1 \times 1$, then $\det(A) = A_{11}$, the single entry of $A$.
		\item If $A$ is $2 \times 2$, then $\det(A) = A_{11}A_{22} - A_{12}A_{21}$.
		\item If $A$ is $n \times n$ for $n > 2$, then

		      \[\det(A) = \sum_{j=1}^{n}(-1)^{i+j}A_{ij}\cdot\det(\tilde{A}_{ij})\]

		      (if the determinant is evaluated by the entries of row $i$ of $A$) or

		      \[\det(A) = \sum_{i=1}^{n}(-1)^{i+j}A_{ij}\cdot\det(\tilde{A}_{ij})\]

		      (if the determinant is evaluated by the entries of column $j$ of $A$), where $\tilde{A}_{ij}$ is the $(n-1) \times (n-1)$ matrix obtained by deleting row $i$ and column $j$ from $A$.
	\end{enumerate}

	In the formulas above, the scalar $(-1)^{i+j}\det(\tilde{A}_{ij})$ is called the \textbf{cofactor} of the row $i$ column $j$ of $A$.
\end{definition}

\begin{definition}[\textbf{Properties of the Determinant}]
	\begin{enumerate}
		\item[]
		\item If $B$ is a matrix obtained by interchanging any two rows or interchanging any two columns of an $n \times n$ matrix $A$, then $\det(B) = -\det(A)$.
		\item If $B$ is a matrix obtained by multiplying each entry of some row or column of an $n \times n$ matrix $A$ by a scalar $k$, then $\det(B) = k\cdot\det(A)$.
		\item If $B$ is a matrix obtained from an $n \times n$ matrix $A$ by adding a multiple of row $i$ to row $j$ or a multiple of column $i$ to column $j$ for $i \neq j$, then $\det(B) = \det(A)$.
	\end{enumerate}
\end{definition}