Chapter 4, Problem 14PT

To determine

To calculate: The determinant of the matrix,

  $\left[\begin{array}{ccc}2& 3& -1\\ 0& 2& 4\\ -2& 5& 6\end{array}\right]$

The possible options are:

  $\begin{array}{l}F-44\\ G44\\ H\frac{1}{44}\\ J-\frac{1}{44}\end{array}$

Answer to Problem 14PT

The value of the determinant is $-44$ . The correct option is F .

Explanation of Solution

Given information:

The matrix is given as:

  $\left[\begin{array}{ccc}2& 3& -1\\ 0& 2& 4\\ -2& 5& 6\end{array}\right]$

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,

  $\left[\begin{array}{ccc}2& 3& -1\\ 0& 2& 4\\ -2& 5& 6\end{array}\right]$

Expand the determinant through the first row,

  $\left|\begin{array}{ccc}2& 3& -1\\ 0& 2& 4\\ -2& 5& 6\end{array}\right|=2\left|\begin{array}{cc}2& 4\\ 5& 6\end{array}\right|-3\left|\begin{array}{cc}0& 4\\ -2& 6\end{array}\right|-1\left|\begin{array}{cc}0& 2\\ -2& 5\end{array}\right|$

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,

  $\begin{array}{c}\left|\begin{array}{ccc}2& 3& -1\\ 0& 2& 4\\ -2& 5& 6\end{array}\right|=2\left|\begin{array}{cc}2& 4\\ 5& 6\end{array}\right|-3\left|\begin{array}{cc}0& 4\\ -2& 6\end{array}\right|-1\left|\begin{array}{cc}0& 2\\ -2& 5\end{array}\right|\\ =2\left(12-20\right)-3\left(8\right)-1\left(4\right)\\ =-16-24-4\\ =-44\end{array}$

Thus, the value of the determinant is $-44$ . The correct option is F .