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 $B_{ij}=A_{ji}$ .
$A = \begin{bmatrix} a & b \newline c & d \newline e & f \end{bmatrix}$
$A^T = \begin{bmatrix} a & c & e \newline b & d & f \newline \end{bmatrix}$