Chapter 16.6, Problem 1WE

To determine

To Find: The value inverse of:

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

Answer to Problem 1WE

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

Explanation of Solution

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

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:

  $\left|\begin{array}{cc}2& 1\\ -1& 2\end{array}\right|=(2)(2)-(1)(-1)=4+1=5$

Determinant is non zero, therefore the inverse exists.

Applying the inverse formula:

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