Chapter 16.5, Problem 18WE

To determine

To show: that if the two row or column of $A$ is multiplied by a scalar $r$ , then $\det A$ is multiplies by $r$ .

Explanation of Solution

Given information:

Let $A$ be any $2\times 2$ matrix..

Calculation:

Let, $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ . Then $\det A$ defined as follows,

  $\begin{array}{c}\det A=A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\\ \det A=ad-bc\end{array}$

Now the $2\times 2$ determinant is multiplied by a scalar $r$ , then

  $Ar=\left[\begin{array}{cc}ar& br\\ c& d\end{array}\right]$

Then its determinant is,

  $\begin{array}{c}Ar=\left[\begin{array}{cc}ar& br\\ c& d\end{array}\right]\\ \det Ar=ar\cdot d-br\cdot c\\ \det Ar=r\left(ad-bc\right)\\ \det Ar=r\det A\end{array}$

Thus, if any row or column of $A$ is multiplied by a scalar $r$ , then $\det Ar=r\det A$ .