73 lines
2.3 KiB
TeX
73 lines
2.3 KiB
TeX
\begin{theorem}[Properties of Inversion]\ \\
|
|
Let $A$ be a square matrix.
|
|
\begin{itemize}
|
|
\item Let $A$ be an invertible matrix, then its inverse is unique.
|
|
\item Cancellation Law: Let $A$ be an invertible matrix
|
|
\subitem $AB_1 = AB_2 \implies B_1 = B_2$
|
|
\subitem $C_1A = C_2A \implies C_1 = C_2$
|
|
\subitem $AB = 0 \implies B = 0, CA = 0 \implies C = 0$ ($A$ is invertible, A cannot be 0)
|
|
\subitem This fails if $A$ is singular
|
|
\item Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$
|
|
\subitem $A$ is invertible $\iff ad - bc \neq 0$
|
|
\subitem $A$ is invertible $A^{-1} = \dfrac{1}{ad - bc} \begin{pmatrix}d & -b \\ -c & a \end{pmatrix}$
|
|
|
|
\end{itemize}
|
|
Let $A$ and $B$ be invertible matrices of same order
|
|
\begin{itemize}
|
|
\item Let $c \neq 0$. Then $cA$ is invertible, $(cA^{-1} = \frac{1}{c}A^{-1}$
|
|
\item $A^T$ is invertible, $(A^T)^{-1} = (A^{-1})^T$
|
|
\item $AB$ is invertible, $(AB)^{-1} = (B^{-1}A^{-1})$
|
|
\end{itemize}
|
|
|
|
Let $A$ be an invertible matrix.
|
|
|
|
\begin{itemize}
|
|
\item $A^{-k} = (A^{-1})^k$
|
|
\item $A^{m+n} = A^mA^n$
|
|
\item $(A^m)^n = A^{mn}$
|
|
\end{itemize}
|
|
|
|
|
|
\end{theorem}
|
|
|
|
\begin{defn}[Elementary Matrices] If it can be obtained from $I$ by performing single elementary row operation
|
|
\begin{itemize}
|
|
\item $cRi, c \neq 0: \begin{pmatrix}
|
|
1 & 0 & 0 & 0 \\
|
|
0 & 1 & 0 & 0 \\
|
|
0 & 0 & c & 0 \\
|
|
0 & 0 & 0 & 1
|
|
\end{pmatrix}(cR_3)$
|
|
\item $R_i \leftrightarrow R_j, i \neq j,: \begin{pmatrix}
|
|
1 & 0 & 0 & 0 \\
|
|
0 & 0 & 0 & 1 \\
|
|
0 & 0 & 1 & 0 \\
|
|
0 & 1 & 0 & 0
|
|
\end{pmatrix}(R_2 \leftrightarrow R_4)$
|
|
\item $R_i + cR_j, i \neq j,: \begin{pmatrix}
|
|
1 & 0 & 0 & 0 \\
|
|
0 & 1 & 0 & c \\
|
|
0 & 0 & 1 & 0 \\
|
|
0 & 0 & 0 & 1
|
|
\end{pmatrix}(R_2 + cR_4)$
|
|
\item Every elementary Matrix is invertible
|
|
\end{itemize}
|
|
\end{defn}
|
|
|
|
$A = \begin{pmatrix}
|
|
a_{11}&a_{12}&a_{13}\\
|
|
a_{21}&a_{22}&a_{23}\\
|
|
a_{31}&a_{32}&a_{33}\\
|
|
a_{41}&a_{42}&a_{43}
|
|
\end{pmatrix}$, $E = \begin{pmatrix}
|
|
1&0&0&0\\
|
|
0&1&0&0\\
|
|
0&0&c&0\\
|
|
0&0&0&1
|
|
\end{pmatrix}(cR_3)$, $EA = \begin{pmatrix}
|
|
a_{11}&a_{12}&a_{13}\\
|
|
a_{21}&a_{22}&a_{23}\\
|
|
ca_{31}&ca_{32}&ca_{33}\\
|
|
a_{41}&a_{42}&a_{43}
|
|
\end{pmatrix}$
|