Chapter 16.6, Problem 2ST

To determine

To Find:Expansion of the determinant about row one:

  $\left|\begin{array}{ccc}2& -2& 0\\ 1& 1& -2\\ 0& -3& 1\end{array}\right|$

Answer to Problem 2ST

-8

Explanation of Solution

Given:

  $\left|\begin{array}{ccc}2& -2& 0\\ 1& 1& -2\\ 0& -3& 1\end{array}\right|$

Formula Used:

1. Determinant of a 3 x 3 matrix:

  $\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|=a.\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|-b.\left|\begin{array}{cc}d& f\\ g& i\end{array}\right|+c.\left|\begin{array}{cc}d& e\\ g& h\end{array}\right|$

2. Determinant of a 2 x 2 matrix:

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

  $\begin{array}{l}\left|\begin{array}{ccc}2& -2& 0\\ 1& 1& -2\\ 0& -3& 1\end{array}\right|=2\left|\begin{array}{cc}1& -2\\ -3& 1\end{array}\right|-(-2)\left|\begin{array}{cc}1& -2\\ 0& 1\end{array}\right|+0\left|\begin{array}{cc}1& 1\\ 0& -3\end{array}\right|\\ \Rightarrow 2(1-6)+2(1)+0\\ \Rightarrow -10+2\\ \Rightarrow -8\end{array}$