Chapter 11.3, Problem 64AYU

To determine

To show: the entries in column 1 equal those in column 3 has the determinant value 0.

Answer to Problem 64AYU

The determinant in which the entries in column 1 equal those in column 3 has the value 0.

Explanation of Solution

Given information: A3 by 3v determinant

Let, consider a determinant whose column 1 & column 3 is equal .

  $=\left|\begin{array}{l}\text{x y z}\\ \text{p q r}\\ \text{x y z}\end{array}\right|$

Therefore , The determinant is

  $\begin{array}{l}=x(qz-ry)-y(pz-rx)+z(py-qx)\\ =qxz-rxy-pyz+rxy+pyzi-qxz\\ (qxz-qxz)+(rxy-rxy)+(pxz-pyz)\\ =0\end{array}$

(proved)

Hence, The determinant in which the entries in column 1 equal those in column 3 has the value 0.