A Few Matrix Applications

Example 2 (Matrix inverse):

$A = \begin{bmatrix}1 & 1 \\ 2 & 3\end{bmatrix}$, $b = \begin{bmatrix}3 \\ 7\end{bmatrix}$

First, we'll compute $A^{-1}$: the inverse of $A$. To do so, we use the matrix inverse formula for a $2 \times 2$ matrix:

$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

where $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$.