Chapter 8.4, Problem 78E

To determine

To verify: The equation,

  $\begin{array}{cc}w& x\\ cw& cx\end{array}=0$

With help of determinants.

Explanation of Solution

Given information:

The equation,

  $\begin{array}{cc}w& x\\ cw& cx\end{array}=0$

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 $\left|A\right|=ps-qr$ .

Calculation:

Consider the equation,

  $\begin{array}{cc}w& x\\ cw& cx\end{array}=0$

Recall that 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 $\left|A\right|=ps-qr$ .

Therefore, determinant of the matrix on the left hand side is given by,

  $\begin{array}{c}\begin{array}{cc}w& x\\ cw& cx\end{array}=\left(w\right)\left(cx\right)-\left(x\right)\left(cw\right)\\ =wcx-xcw\\ =wcx-wcx\\ =0\end{array}$

Since, the left hand side is equal to right hand side, the equation is verified.

Thus, the equation, $\begin{array}{cc}w& x\\ cw& cx\end{array}=0$ is verified with help of determinants.