Chapter 11.3, Problem 43AYU

In problems 43-50, use properties of determinants to find the value of each determinant if it is known that $\left[\begin{array}{c}x\\ u\\ 1\end{array}\begin{array}{c}y\\ v\\ 2\end{array}\begin{array}{c}z\\ w\\ 3\end{array}\right]=4$

$\left[\begin{array}{c}1\\ u\\ x\end{array}\begin{array}{c}2\\ v\\ y\end{array}\begin{array}{c}3\\ w\\ z\end{array}\right]$

To determine

To find: The value of the determinant: $\left|\begin{array}{ccc}1& 2& 3\\ u& v& w\\ x& y& z\end{array}\right|$.

Answer to Problem 43AYU

$-4$

Explanation of Solution

Given:

$\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4$

Calculation:

Interchanging the first and the third row we get,

$\left|\begin{array}{ccc}1& 2& 3\\ u& v& w\\ x& y& z\end{array}\right|={\left(-1\right)}^3\cdot \left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=-4$