Chapter 16.8, Problem 11WE

To determine

To calculate: The value of the determinant,

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

Answer to Problem 11WE

The value of the determinant is $0$ .

Explanation of Solution

Given information:

The determinant:

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

Formula used:

The value of new determinant is same as the original one if all entries of either a row or a column are multiplied by a real number k , and the resultant products are added to corresponding entries of either another row or another column.

Calculation:

Consider the determinant

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

Recall that the value of new determinant is same as the original one if all entries of either a row or a column are multiplied by a real number k , and the resultant products are added to corresponding entries of either another row or another column.

Add the corresponding elements of column two to column one.

  $\left|\begin{array}{cccc}1& -1& -1& 1\\ 1& 1& 1& -1\\ -1& -1& 1& 1\\ -1& 1& -1& -1\end{array}\right|=\left|\begin{array}{cccc}0& -1& -1& 1\\ 2& 1& 1& -1\\ -2& -1& 1& 1\\ 0& 1& -1& -1\end{array}\right|$

Add the corresponding elements of column four to column two.

  $\begin{array}{c}\left|\begin{array}{cccc}1& -1& -1& 1\\ 1& 1& 1& -1\\ -1& -1& 1& 1\\ -1& 1& -1& -1\end{array}\right|=\left|\begin{array}{cccc}0& -1& -1& 1\\ 2& 1& 1& -1\\ -2& -1& 1& 1\\ 0& 1& -1& -1\end{array}\right|\\ =\left|\begin{array}{cccc}0& 0& -1& 1\\ 2& 0& 1& -1\\ -2& 0& 1& 1\\ 0& 0& -1& -1\end{array}\right|\end{array}$

Observe that all entries of second column are zero.

Recall that if either an entire row or an entire column is 0 then the value of determinant is 0.

Therefore,

  $\begin{array}{c}\left|\begin{array}{cccc}1& -1& -1& 1\\ 1& 1& 1& -1\\ -1& -1& 1& 1\\ -1& 1& -1& -1\end{array}\right|=\left|\begin{array}{cccc}0& -1& -1& 1\\ 2& 1& 1& -1\\ -2& -1& 1& 1\\ 0& 1& -1& -1\end{array}\right|\\ =\left|\begin{array}{cccc}0& 0& -1& 1\\ 2& 0& 1& -1\\ -2& 0& 1& 1\\ 0& 0& -1& -1\end{array}\right|\\ =0\end{array}$

Thus, the value of the determinant is $0$ .