Chapter 8.4, Problem 88E

To determine

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

  $\begin{array}{cc}3x^2& -3y^2\\ 1& 1\end{array}$

Answer to Problem 88E

The value of the determinant is $3x^2+3y^2$ .

Explanation of Solution

Given information:

The matrix is given as:

  $\begin{array}{cc}3x^2& -3y^2\\ 1& 1\end{array}$

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,

  $\begin{array}{cc}3x^2& -3y^2\\ 1& 1\end{array}$

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,

  $\begin{array}{c}\begin{array}{cc}3x^2& -3y^2\\ 1& 1\end{array}=\left(3x^2\right)\left(1\right)-\left(-3y^2\right)\left(1\right)\\ =3x^2+3y^2\end{array}$

Thus, the value of the determinant is $3x^2+3y^2$ .