Matrix Inverse & Transpose

Matrix inverse: If $A$ is an $m\times m$ matrix, and if it has an inverse, then $A\times A^{-1}=A^{-1}\times A=I$
$A=\begin{bmatrix}
a & b \newline
c & d \newline
\end{bmatrix} $
$A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}
d & -b \newline
-c & a \newline
\end{bmatrix}$
Note: Matrices that do not have an inverse are singular or degenerate.

Matrix transpose: Let $A$ be an $m\times n$ matrix, and let $B=A^T$. then B is an $n\times n$ matrix and $B_{ij}=A_{ji}$.
$A =
\begin{bmatrix}
a & b \newline
c & d \newline
e & f
\end{bmatrix}$
$A^T =
\begin{bmatrix}
a & c & e \newline
b & d & f \newline
\end{bmatrix}$