# How do you find the inverse of [(9,13), (27,36)]?

Jul 22, 2017

${A}^{- 1} = \left[\begin{matrix}- \frac{4}{3} & \frac{13}{27} \\ 1 & - \frac{1}{3}\end{matrix}\right]$

#### Explanation:

Let's say $A = \left[\begin{matrix}9 & 13 \\ 27 & 36\end{matrix}\right]$

The inverse is then : ${A}^{- 1} = \frac{1}{\det \left(A\right)} a \mathrm{dj} \left(A\right)$

$\frac{1}{\det \left(A\right)} = \frac{1}{9 \cdot 36 - 13 \cdot 27} = - \frac{1}{27}$

$a \mathrm{dj} \left(A\right) = \left[\begin{matrix}36 & - 13 \\ - 27 & 9\end{matrix}\right]$

So ${A}^{- 1} = - \frac{1}{27} \left[\begin{matrix}36 & - 13 \\ - 27 & 9\end{matrix}\right] = \left[\begin{matrix}- \frac{36}{27} & \frac{13}{27} \\ \frac{27}{27} & - \frac{9}{27}\end{matrix}\right] =$

$\left[\begin{matrix}- \frac{4}{3} & \frac{13}{27} \\ 1 & - \frac{1}{3}\end{matrix}\right]$

Jul 22, 2017

The inverse is $= - \frac{1}{27} \left(\begin{matrix}36 & - 13 \\ - 27 & 9\end{matrix}\right)$

#### Explanation:

The inverse of the matrix $A = \left(\begin{matrix}a & b \\ c & d\end{matrix}\right)$ is

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

Let $A = \left(\begin{matrix}9 & 13 \\ 27 & 36\end{matrix}\right)$

The determinant of $A$ is

$\det A = | \left(\begin{matrix}9 & 13 \\ 27 & 36\end{matrix}\right) | = 36 \cdot 9 - 27 \cdot 13 = - 27$

As $\det A \ne 0$, the matrix $A$ is invertible

${A}^{-} 1 = - \frac{1}{27} \left(\begin{matrix}36 & - 13 \\ - 27 & 9\end{matrix}\right)$

Verification

$A \cdot {A}^{-} 1 = \left(\begin{matrix}9 & 13 \\ 27 & 36\end{matrix}\right) \cdot \left(- \frac{1}{27}\right) \left(\begin{matrix}36 & - 13 \\ - 27 & 9\end{matrix}\right) = \left(\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right)$

Jul 22, 2017

${A}^{-} 1 = \left(\begin{matrix}- \frac{4}{3} & \frac{13}{27} \\ 1 & - \frac{1}{3}\end{matrix}\right)$

#### Explanation:

$\text{given } A = \left(\begin{matrix}a & b \\ c & d\end{matrix}\right)$

$\text{then the inverse "A^-1" is}$

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

$\det A = a d - b c$

$\Rightarrow \det A = \left(9 \times 36\right) - \left(13 \times 27\right) = - 27$

$\Rightarrow {A}^{-} 1 = - \frac{1}{27} \left(\begin{matrix}36 & - 13 \\ - 27 & 9\end{matrix}\right)$

$\textcolor{w h i t e}{r A e e {A}^{-} 1} = \left(\begin{matrix}- \frac{4}{3} & \frac{13}{27} \\ 1 & - \frac{1}{3}\end{matrix}\right)$