Chapter 8.4, Problem 87E

To determine

To calculate: The determinant where the entries are the functions,

  $\begin{array}{cc}4u& -1\\ -1& 2v\end{array}$

Answer to Problem 87E

The value of the determinantis $8uv-1$ .

Explanation of Solution

Given information:

The matrix is given as:

  $\begin{array}{cc}4u& -1\\ -1& 2v\end{array}$

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 matrix,

  $\begin{array}{cc}4u& -1\\ -1& 2v\end{array}$

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 is given by,

  $\begin{array}{c}\begin{array}{cc}4u& -1\\ -1& 2v\end{array}=\left(4u\right)\left(2v\right)-\left(-1\right)\left(-1\right)\\ =8uv-1\end{array}$

Thus, the value of the determinant is $8uv-1$ .