Q.33 - Precalculus Enhanced with Graphing Utilities

Question

In Problems $32$and$33$, use properties of determinants to find the value of each determinant if it is known that $\left|\begin{array}{cc}x& y\\ a& b\end{array}\right|=8$

$\left|\begin{array}{cc}y& x\\ b& a\end{array}\right|$

Step-by-Step Solution

Answer

If$\left|\begin{array}{cc}x& y\\ a& b\end{array}\right|=8$, then the value of the determinant $\left|\begin{array}{cc}y& x\\ b& a\end{array}\right|$is$-8$.

Step 1. Given data

The given determinant is  

$\left|\begin{array}{cc}x& y\\ a& b\end{array}\right|=8$

The determinant whose value need to find out is 

$\left|\begin{array}{cc}y& x\\ b& a\end{array}\right|$

Step 2. Value of determinant

According to properties of determinants, if any column of determinants is swiped with another column then the value of determinants will be negated.

$\left|\begin{array}{cc}y& x\\ b& a\end{array}\right|$is obtained by swapping the first column with the second column of$\left|\begin{array}{cc}x& y\\ a& b\end{array}\right|$

so the value of the determinant $\left|\begin{array}{cc}y& x\\ b& a\end{array}\right|$ is negative of the value of the determinant$\left|\begin{array}{cc}x& y\\ a& b\end{array}\right|$

So$$\begin{gathered} \left|\begin{array}{cc}x& y\\ a& b\end{array}\right|=8 \\ \left|\begin{array}{cc}y& x\\ b& a\end{array}\right|=-8 \end{gathered}$$