77 lines
2.4 KiB
TeX
77 lines
2.4 KiB
TeX
\section{Determinants of Order 2}
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\begin{definition}
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\hfill\\
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If
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\[A = \begin{pmatrix}
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a & b \\
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c & d
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\end{pmatrix}\]
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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 $ad-bc$.
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\end{definition}
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\begin{theorem}
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\hfill\\
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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
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\[\det \begin{pmatrix}
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u + kv \\
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w
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\end{pmatrix} = \det\begin{pmatrix}
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u \\ w
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\end{pmatrix} + k\det\begin{pmatrix}
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v \\ w
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\end{pmatrix}\]
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and
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\[\det\begin{pmatrix}
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w \\ u + kv
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\end{pmatrix} = \det\begin{pmatrix}
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w \\ u
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\end{pmatrix} + k \det \begin{pmatrix}
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w \\ v
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\end{pmatrix}.\]
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\end{theorem}
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\begin{theorem}\label{Theorem 4.2}
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\hfill\\
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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
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\[A^{-1} = \frac{1}{\det(A)}\begin{pmatrix}
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A_{22} & -A_{12} \\
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-A_{21} & A_{11}
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\end{pmatrix}.\]
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\end{theorem}
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\begin{definition}
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\hfill\\
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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.
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\end{definition}
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\begin{definition}
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\hfill\\
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If $\beta = \{u,v\}$ is an ordered basis for $\R^2$, we define the \textbf{orientation} of $\beta$ to be the real number
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\[O\begin{pmatrix}
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u \\ v
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\end{pmatrix} = \frac{\det\begin{pmatrix}
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u \\ v
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\end{pmatrix}}{\abs{\det\begin{pmatrix}
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u \\ v
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\end{pmatrix}}}\]
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(The denominator of this fraction is nonzero by \autoref{Theorem 4.2}).
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\end{definition}
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\begin{definition}
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\hfill\\
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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.
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\end{definition}
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\begin{definition}
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\hfill\\
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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 $v$}.
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\end{definition}
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