Question

Show that the function

$\mathit{F}\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& j\end{array}\right]\mathbf{=}\mathit{b}\mathit{f}\mathit{g}$

is linear in all three columns and in all three rows. See Example 6. Is F alternating on the columns? See Example 4.

Short answer

Therefore, function F is linear in all rows and columns, but it's not alternating on the columns.

Step-by-step solution

Given

Given matrix,

$F\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& j\end{array}\right]=bfg$

To show the function

Given,

$F\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& j\end{array}\right]=bfg$

Contains one factor from each row and from each column, therefore, it's linear in all three rows and all three columns.

On the other hand, we can see that, for example,

role="math" localid="1659505275434" $F\left(\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]\right)=1$, but if we switch the second and third column of this matrix, we have

$F\left(\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]\right)=1$, which means that F is not alternating on the columns.

Therefore, function F is linear in all rows and columns, but it's not alternating on the