Q45. - Algebra 2

Question

Find a third order determinant in which no element is 0, but for which the determinant is 0.

Step-by-Step Solution

Answer

Thus, a third order determinant in which no element is 0, but for which the determinant is 0 is $\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|$

Step 1 - Write a third order determinant

A third order determinant is determinant of a square matrix of order 3.

It can be simplified as shown below

 $\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|=a\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|-b\left|\begin{array}{cc}d& f\\ g& i\end{array}\right|+c\left|\begin{array}{cc}d& e\\ g& h\end{array}\right|$

Step 2 - Write a third order determinant in which no element is 0

Substitute 1 for all elements in a third order determinant

 $\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|$

Step 3 - Simplify 1 1 1 1 1 1 1 1 1

To evaluate this determinant expand by using minors

$\begin{array}{c}\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|=1\left|\begin{array}{cc}1& 1\\ 1& 1\end{array}\right|-1\left|\begin{array}{cc}1& 1\\ 1& 1\end{array}\right|+1\left|\begin{array}{cc}1& 1\\ 1& 1\end{array}\right|\\ =1\left(1-1\right)-1\left(1-1\right)+1\left(1-1\right)\\ =0-0+0\\ =0\end{array}$