# Given A=((-1, 2), (3, 4)) and B=((-4, 3), (5, -2)), how do you find A^-1?

May 28, 2016

${A}^{- 1} = \left(\begin{matrix}- \frac{2}{5} & \frac{1}{5} \\ \frac{3}{10} & \frac{1}{10}\end{matrix}\right)$.

#### Explanation:

There is a rule to calculate the inverse of a $2 \setminus \times 2$ matrix.
If we have a matrix $M$ with components
$M = \left(\begin{matrix}a & b \\ c & d\end{matrix}\right)$ the inverse is
${M}^{- 1} = \frac{1}{a d - b c} \left(\begin{matrix}d & - b \\ - c & a\end{matrix}\right)$.

We can apply this to the matrix $A$ and obtain

${A}^{- 1} = \frac{1}{- 1 \cdot 4 - 2 \cdot 3} \left(\begin{matrix}4 & - 2 \\ - 3 & - 1\end{matrix}\right)$
$= \frac{1}{- 10} \left(\begin{matrix}4 & - 2 \\ - 3 & - 1\end{matrix}\right)$
$= \left(\begin{matrix}- \frac{4}{10} & \frac{2}{10} \\ \frac{3}{10} & \frac{1}{10}\end{matrix}\right)$
$= \left(\begin{matrix}- \frac{2}{5} & \frac{1}{5} \\ \frac{3}{10} & \frac{1}{10}\end{matrix}\right)$.

Unfortunately I do not understand what should be the role of B because to invert a matrix you don't need another matrix.