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}