# How do you find the inverse of A=((9, -5), (-7, 4))?

Feb 18, 2016

$\left(\begin{matrix}4 & 5 \\ 7 & 9\end{matrix}\right)$

#### Explanation:

For a 2 x 2 matrix , the inverse may be found as follows:

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

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

ad-bc is the determinant of the matrix and it's value determines whether the matrix has an inverse or not
If (ad - bc ) = 0 then an inverse does not exist.

in this question a=9 , b=-5 , c = -7 and d=4

ad-bc $= \left(9 \times 4\right) - \left(- 5 \times \left(- 7\right)\right) = 36 - 35 = 1$
hence an inverse exists.

${A}^{-} 1 = \left(\begin{matrix}4 & 5 \\ 7 & 9\end{matrix}\right)$

The inverse of a matrix may also be found using
$\textcolor{b l u e}{\text{ Gaussian Elimination }}$
but is usually used in higher order matrices $3 \times 3 \text{ and above}$