Chapter 16.7, Problem 9WE

To determine

To Find:Expansion of the determinant:

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

Answer to Problem 9WE

-4

Explanation of Solution

Given:

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

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:

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

  $\begin{array}{l}\left|\begin{array}{ccc}1& 0& -1\\ 0& 1& 2\\ -1& 2& 1\end{array}\right|=1\left|\begin{array}{cc}1& 2\\ 2& 1\end{array}\right|-0\left|\begin{array}{cc}0& 2\\ -1& 1\end{array}\right|+(-1)\left|\begin{array}{cc}0& 1\\ -1& 2\end{array}\right|\\ \Rightarrow 1(1-4)-0-1(0-(-1))\\ \Rightarrow -3-1\\ \Rightarrow -4\end{array}$