Chapter 16.6, Problem 2WE

To determine

To Find: The value inverse of:

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

Answer to Problem 2WE

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

Explanation of Solution

Given: $\left[\begin{array}{cc}5& 3\\ 2& 1\end{array}\right]$

Formula Used:

1. Determinant formula:

  $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-cb$

2. Inverse formula:

  ${\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Finding the determinant:

  $\left|\begin{array}{cc}5& 3\\ 2& 1\end{array}\right|=(5)(1)-(2)(3)=5-6=-1$

Determinant is non zero, therefore the inverse exists.

Applying the inverse formula:

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