Chapter 16.7, Problem 10WE

To determine

To Find:Expansion of the determinant:

  $\left|\begin{array}{ccc}0& 2& 1\\ 4& 0& -2\\ -1& 3& 1\end{array}\right|$

Answer to Problem 10WE

8

Explanation of Solution

Given:

  $\left|\begin{array}{ccc}0& 2& 1\\ 4& 0& -2\\ -1& 3& 1\end{array}\right|$

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:

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

  $\begin{array}{l}\left|\begin{array}{ccc}0& 2& 1\\ 4& 0& -2\\ -1& 3& 1\end{array}\right|=0\left|\begin{array}{cc}0& -2\\ 3& 1\end{array}\right|-2\left|\begin{array}{cc}4& -2\\ -1& 1\end{array}\right|+1\left|\begin{array}{cc}4& 0\\ -1& 3\end{array}\right|\\ \Rightarrow 0-2(4-2)+1(12-0)\\ \Rightarrow 12-4\\ \Rightarrow 8\end{array}$