Chapter 11.3, Problem 62AYU

To determine

To show: 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.

Explanation of Solution

Given information:

The determinant of a 3 by 3 matrix.

Formula used:

For the matrix of order $2\times 2$ ,

  $A=\left[\begin{array}{ll}p\hfill & q\hfill \\ r\hfill & s\hfill \end{array}\right]$

The determinant is given as $\left|A\right|=ps-qr$ .

Proof:

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.