Chapter 11, Problem 28RE

To determine

To find: The value of determinant.

Answer to Problem 28RE

The value of determinant is $-100$ .

Explanation of Solution

Given information:

The determinant is $\left|\begin{array}{ccc}2& 1& -3\\ 5& 0& 1\\ 2& 6& 0\end{array}\right|$ .

Calculation:

Consider the determinant.

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

The formula for solving a $3\times 3$ determinant is given by,

  $\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|=a_{12}\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|-a_{12}\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|-a_{13}\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$

The formula for solving the determinant is,

  $\begin{array}{c}D=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\\ =ad-bc\end{array}$

Solve the determinant using above formula.

  $\begin{array}{c}\left|\left|\begin{array}{ccc}2& 1& -3\\ 5& 0& 1\\ 2& 6& 0\end{array}\right|\right|=2\left|\begin{array}{cc}0& 1\\ 6& 0\end{array}\right|-1\left|\begin{array}{cc}5& 1\\ 2& 0\end{array}\right|+\left(-3\right)\left|\begin{array}{cc}5& 0\\ 0& 6\end{array}\right|\\ =2\left(0\right)-1\left[0-2\right]-3\left(30-0\right)\\ =-100\end{array}$

Therefore, the value of determinant is $-100$ .