Chapter 16.7, Problem 4WE

To determine

To Find:Expansion of the determinant about column 3:

  $\left|\begin{array}{ccc}4& 0& -5\\ 9& 1& 11\\ -4& 0& 3\end{array}\right|$

Answer to Problem 4WE

-8

Explanation of Solution

Given:

  $\left|\begin{array}{ccc}4& 0& -5\\ 9& 1& 11\\ -4& 0& 3\end{array}\right|$

Formula Used:

1. Determinant of a 2 x 2 matrix:

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

While expanding about column 2, the sequence of signs goes like: Negative, Positive, Negative.

  $\begin{array}{l}\left|\begin{array}{ccc}4& 0& -5\\ 9& 1& 11\\ -4& 0& 3\end{array}\right|=-0\left|\begin{array}{cc}9& 11\\ -4& 3\end{array}\right|+1\left|\begin{array}{cc}4& -5\\ -4& 3\end{array}\right|-0\left|\begin{array}{cc}4& -5\\ 9& 11\end{array}\right|\\ \Rightarrow 0+1(12-20)+0\\ \Rightarrow 12-20\\ \Rightarrow -8\end{array}$