Question

In Problems $32$and$33$, use properties of determinants to find the value of each determinant if it is known thatrole="math" localid="1646935214028" $\left|\begin{array}{cc}x& y\\ a& b\end{array}\right|=8$

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

Short 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-by-step solution

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