80 lines
2.5 KiB
TeX
80 lines
2.5 KiB
TeX
\section{Determinants of Order 2}
|
|
|
|
\begin{definition}
|
|
\hfill\\
|
|
If
|
|
|
|
\[A = \begin{pmatrix}
|
|
a & b \\
|
|
c & d
|
|
\end{pmatrix}\]
|
|
is a $2 \times 2$ matrix with entries from a field $\F$, then we define the \textbf{determinant} of $A$, denoted $\det(A)$ or $|A|$, to be the scalar $ad-bc$.
|
|
\end{definition}
|
|
|
|
\begin{theorem}
|
|
\hfill\\
|
|
The function $\det: M_{2 \times 2}(\F) \to \F$ is a linear function of each row of a $2 \times 2$ matrix when the other row is held fixed. That is, if $u$, $v$ and $w$ are in $\F^2$ and $k$ is a scalar, then
|
|
|
|
\[\det \begin{pmatrix}
|
|
u + kv \\
|
|
w
|
|
\end{pmatrix} = \det\begin{pmatrix}
|
|
u \\ w
|
|
\end{pmatrix} + k\det\begin{pmatrix}
|
|
v \\ w
|
|
\end{pmatrix}\]
|
|
|
|
and
|
|
|
|
\[\det\begin{pmatrix}
|
|
w \\ u + kv
|
|
\end{pmatrix} = \det\begin{pmatrix}
|
|
w \\ u
|
|
\end{pmatrix} + k \det \begin{pmatrix}
|
|
w \\ v
|
|
\end{pmatrix}.\]
|
|
\end{theorem}
|
|
|
|
\begin{theorem}\label{Theorem 4.2}
|
|
\hfill\\
|
|
Let $A \in M_{2 \times 2}(\F)$. Then the determinant of $A$ is nonzero if and only if $A$ is invertible. Moreover, if $A$ is invertible, then
|
|
|
|
\[A^{-1} = \frac{1}{\det(A)}\begin{pmatrix}
|
|
A_{22} & -A_{12} \\
|
|
-A_{21} & A_{11}
|
|
\end{pmatrix}.\]
|
|
\end{theorem}
|
|
|
|
\begin{definition}
|
|
\hfill\\
|
|
By the \textbf{angle} between two vectors in $\R^2$, we mean the angle with measure $\theta$ ($0 \leq \theta < \pi$) that is formed by the vectors having the same magnitude and direction as the given vectors by emanating from the origin.
|
|
\end{definition}
|
|
|
|
\subsection*{The Area of a Parallelogram}
|
|
\addcontentsline{toc}{subsection}{The Area of a Parallelogram}
|
|
|
|
\begin{definition}
|
|
\hfill\\
|
|
If $\beta = \{u,v\}$ is an ordered basis for $\R^2$, we define the \textbf{orientation} of $\beta$ to be the real number
|
|
|
|
\[O\begin{pmatrix}
|
|
u \\ v
|
|
\end{pmatrix} = \frac{\det\begin{pmatrix}
|
|
u \\ v
|
|
\end{pmatrix}}{\abs{\det\begin{pmatrix}
|
|
u \\ v
|
|
\end{pmatrix}}}\]
|
|
|
|
(The denominator of this fraction is nonzero by \autoref{Theorem 4.2}).
|
|
\end{definition}
|
|
|
|
\begin{definition}
|
|
\hfill\\
|
|
A coordinate system $\{u, v\}$ is called \textbf{right-handed} if $u$ can be rotated in a counterclockwise direction through an angle $\theta$ ($0 < \theta < \pi$) to coincide with $v$. Otherwise, $\{u ,v\}$ is called a \textbf{left-handed} system.
|
|
\end{definition}
|
|
|
|
\begin{definition}
|
|
\hfill\\
|
|
Any ordered set $\{u, v\}$ in $\R^2$ determines a parallelogram in the following manner. Regarding $u$ and $v$ as arrows emanating from the origin of $\R^2$, we call the parallelogram having $u$ and $v$ as adjacent sides the \textbf{parallelogram determined by $u$ and $v$}.
|
|
\end{definition}
|