Chapter 16, Problem 13CT

To determine

To find: The value of the given determinant.

Answer to Problem 13CT

The value of the determinant is $0$ .

Explanation of Solution

Given information:

The given determinant is

  $\left|\begin{array}{ccc}10& 5& -5\\ 16& 8& -8\\ 3& 0& -4\end{array}\right|$ .

Calculation:

If each element in one row or one column of a determinant is multiplied by a real number $k$ , then the determinant is multiplied by $k$ .

Factor out $5$ from the elements in the first row,

  $\left|\begin{array}{ccc}10& 5& -5\\ 16& 8& -8\\ 3& 0& -4\end{array}\right|=5\left|\begin{array}{ccc}2& 1& -1\\ 16& 8& -8\\ 3& 0& -4\end{array}\right|$

Factor out $5$ from the elements in the first row,

  $\begin{array}{c}5\left|\begin{array}{ccc}2& 1& -1\\ 16& 8& -8\\ 3& 0& -4\end{array}\right|=5\cdot 8\left|\begin{array}{ccc}2& 1& -1\\ 2& 1& -1\\ 3& 0& -4\end{array}\right|\\ =40\left|\begin{array}{ccc}2& 1& -1\\ 2& 1& -1\\ 3& 0& -4\end{array}\right|\end{array}$

From the above calculation we note that in the above determinant the corresponding element in the first row and the second row are the same.

If two row or two columns of a determinant have corresponding element that are equal, then the determinant is equal to zero.

Therefore, the value of the determinant is $0$ .