Chapter 16.7, Problem 13WE

To determine

To Find: $x$ =?

Answer to Problem 13WE

1

Explanation of Solution

Given:

  $\left|\begin{array}{ccc}2& 1& 1\\ 1& 2& 1\\ x& 1& 2\end{array}\right|=4$

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:

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

  $\begin{array}{l}\left|\begin{array}{ccc}2& 1& 1\\ 1& 2& 1\\ x& 1& 2\end{array}\right|=2\left|\begin{array}{cc}2& 1\\ 1& 2\end{array}\right|-1\left|\begin{array}{cc}1& 1\\ x& 2\end{array}\right|+1\left|\begin{array}{cc}1& 2\\ x& 1\end{array}\right|\\ \Rightarrow 2(4-1)-1(2-x)+1(1-2x)=6-2+x+1-2x=4\\ \Rightarrow -x=-1\\ \Rightarrow x=1\end{array}$