Chapter 11.3, Problem 5AYU

To determine

To explain: Whether the statement “The value of a determinant remains unchanged if any two rows or any two columns are interchanged.” is true or false.

Answer to Problem 5AYU

The statement “The value of a determinant remains unchanged if any two rows or any two columns are interchanged.” is false.

Explanation of Solution

Given information:

The statement “The value of a determinant remains unchanged if any two rows or any two columns are interchanged.”

Consider the provided statement “The value of a determinant remains unchanged if any two rows or any two columns are interchanged.”

The statement is not because if any two rows or any two columns are interchanged the sign of determinant changes.

Consider a 3 by 3 matrix say, $\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$ .

Now, denote the determinant of the above matrix as, $D_1=\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|$ .

Compute it,

  $\begin{array}{c}{D}_1=a\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|-b\left|\begin{array}{cc}d& f\\ g& i\end{array}\right|+c\left|\begin{array}{cc}d& e\\ g& h\end{array}\right|\\ =a\left(ei-fh\right)-b\left(di-fg\right)+c\left(dh-eg\right)\\ =aei-afh-bdi+bfg+cdh-ceg\end{array}$

Now, interchange the column 1 and 3 of the original matrix, so new matrix is $\left(\begin{array}{ccc}c& b& a\\ f& e& d\\ i& h& g\end{array}\right)$

Now, denote the determinant of the above matrix as, $D_2=\left|\begin{array}{ccc}c& b& a\\ f& e& d\\ i& h& g\end{array}\right|$ .

Compute it,

  $\begin{array}{c}{D}_2=c\left|\begin{array}{cc}e& d\\ h& g\end{array}\right|-b\left|\begin{array}{cc}f& d\\ i& g\end{array}\right|+a\left|\begin{array}{cc}f& e\\ i& h\end{array}\right|\\ =c\left(eg-dh\right)-b\left(fg-di\right)+a\left(fh-ei\right)\\ =ceg-cdh-bfg+bdi+afh-aei\\ =-\left(-ceg+cdh+bfg-bdi-afh+aei\right)\end{array}$

Observe that, $D_2=-D_1$ .

Hence, it is shown that when the first and third column of a 3 by 3 determinant are interchanged the value of the new determinant is $-1$ times the value of the original determinant.

Thus, the statement “The value of a determinant remains unchanged if any two rows or any two columns are interchanged.” is false.