Question

Find the inverse of each matrix, if it exists.

$\left[\begin{array}{cc}4& -5\\ 2& -1\end{array}\right]$

Short answer

The inverse of the given matrix is$\left[\begin{array}{cc}-\frac{1}{6}& \frac{5}{6}\\ -\frac{1}{3}& \frac{2}{3}\end{array}\right]$.

Step-by-step solution

Step 1. Given Information.

Given, to find the inverse of the matrix $\left[\begin{array}{cc}4& -5\\ 2& -1\end{array}\right]$.

Step 2. Calculation.

The inverse of matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$, where$ad-bc\ne 0$:

Here,$a=4,b=-5,c=2,d=-1$.

Checking if the inverse exists:

$\begin{array}{l}ad-bc=\left(4\right)\left(-1\right)-\left(-5\right)\left(2\right)\\ ad-bc=-4+10\\ ad-bc=6\end{array}$

Hence, the inverse exists.

The inverse of the given matrix is:

$\begin{array}{l}{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\\ A^{-1}=\frac{1}{\left(4\right)\left(-1\right)-\left(-5\right)\left(2\right)}\left[\begin{array}{cc}-1& -\left(-5\right)\\ -\left(2\right)& 4\end{array}\right]\\ A^{-1}=\frac{1}{-4+10}\left[\begin{array}{cc}-1& 5\\ -2& 4\end{array}\right]\\ A^{-1}=\frac{1}{6}\left[\begin{array}{cc}-1& 5\\ -2& 4\end{array}\right]\\ A^{-1}=\left[\begin{array}{cc}-\frac{1}{6}& \frac{5}{6}\\ -\frac{1}{3}& \frac{2}{3}\end{array}\right]\end{array}$

Step 3. Conclusion.

Hence, the inverse of the given matrix is $\left[\begin{array}{cc}-\frac{1}{6}& \frac{5}{6}\\ -\frac{1}{3}& \frac{2}{3}\end{array}\right]$.