Wrote out chapters 2-4

chapter-4/summary-important-facts-about-determinants.tex

@@ -1 +1,31 @@
 \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}