Chapter 16.6, Problem 4WE

To determine

To Find: The value of inverse of matrix:

  $\left[\begin{array}{cc}-3& 1\\ 4& -2\end{array}\right]$

Answer to Problem 4WE

  $\left[\begin{array}{cc}-1& -0.5\\ -2& -1.5\end{array}\right]$

Explanation of Solution

Given: $\left[\begin{array}{cc}-3& 1\\ 4& -2\end{array}\right]$

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:

  $\left|\begin{array}{cc}-3& 1\\ 4& -2\end{array}\right|=(-3)(-2)-(4)(1)=6-4=2$

Determinant is non zero, therefore the inverse exists.

Applying the inverse formula:

  ${\left[\begin{array}{cc}-3& 1\\ 4& -2\end{array}\right]}^{-1}=\frac{1}{2}\left[\begin{array}{cc}-2& -1\\ -4& -3\end{array}\right]=\left[\begin{array}{cc}-1& -0.5\\ -2& -1.5\end{array}\right]$