Q52. - Algebra 2

Question

Find the inverse of each matrix, if it exists.

 

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

Step-by-Step Solution

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 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  $\left[\begin{array}{cc}-\frac{1}{6}& \frac{5}{6}\\ -\frac{1}{3}& \frac{2}{3}\end{array}\right]$.