Chapter 16.2, Problem 23WE

To determine

To prove: these properties for all $2\times 2$ matrices.

Answer to Problem 23WE

Explanation of Solution

Given information:

Let $p$ and $q$ be scalars and let $A,B$ and $C$ be $2\times 2$ matrices.

Given,

  $pA=Ap$

Calculation: Consider two $2\times 2$ matrices $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

Now, solve the equation $pA=Ap$ as shown,

At first solve $pA$ ,

  $pA=\left[\begin{array}{cc}pa& pb\\ pc& pd\end{array}\right]$

Now, solve $Ap$ ,

  $Ap=\left[\begin{array}{cc}pa& pb\\ pc& pd\end{array}\right]$

Hence, $pA=Ap$ .