Chapter 7.7, Problem 82E

To determine

To explain: why the determinant of the matrix is equal to zero.

Answer to Problem 82E

In given matrices third row having all zero elements so directly write value of its determinant equal to 0 without calculating.

Explanation of Solution

Given information: Given matrix is , $\left(\begin{array}{ccc}2& -4& 5\\ 1& -2& 3\\ 0& 0& 0\end{array}\right)$ .

Formula used:

Leibniz formula;

The Leibniz formula for the determinant of a 3 × 3 matrix:

  $\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)=aei+bfg+cdh-ceg-bdi-afh.$

Calculation:

Given matrix is , $\left(\begin{array}{ccc}2& -4& 5\\ 1& -2& 3\\ 0& 0& 0\end{array}\right)$ .

  $\begin{array}{l}\left(\begin{array}{ccc}2& -4& 5\\ 1& -2& 3\\ 0& 0& 0\end{array}\right)=(2)(-2)(0)+(-4)(3)(0)+(5)(1)(0)-(5)(1)(0)-(-4)(1)(0)-(2)(3)(0).\\ \left(\begin{array}{ccc}2& -4& 5\\ 1& -2& 3\\ 0& 0& 0\end{array}\right)=0+0+0-0-0-0=0.\end{array}$ .

So, it is one of the important properties of determinant of matrices. If, any matrix consisting of a row or column having all zero elements then directly write value of its determinant equal to 0.

In given matrices third row having all zero elements so directly write value of its determinant equal to 0 without calculating.