\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}

$A = \begin{pmatrix}
  a_{11}&a_{12}&a_{13}\\ 
  a_{21}&a_{22}&a_{23}\\ 
  a_{31}&a_{32}&a_{33}\\ 
  a_{41}&a_{42}&a_{43}
\end{pmatrix}$, $E = \begin{pmatrix}
  1&0&0&0\\ 
  0&1&0&0\\ 
  0&0&c&0\\ 
  0&0&0&1
\end{pmatrix}(cR_3)$, $EA = \begin{pmatrix}
  a_{11}&a_{12}&a_{13}\\ 
  a_{21}&a_{22}&a_{23}\\ 
  ca_{31}&ca_{32}&ca_{33}\\ 
  a_{41}&a_{42}&a_{43}
\end{pmatrix}$
