Chapter 16.8, Problem 3WE

To determine

To calculate: The value of the determinant,

  $\left|\begin{array}{ccc}2& -1& 1\\ 1& 3& -3\\ 7& -8& 8\end{array}\right|$

Answer to Problem 3WE

The value of the determinant is $0$ .

Explanation of Solution

Given information:

The determinant:

  $\left|\begin{array}{ccc}2& -1& 1\\ 1& 3& -3\\ 7& -8& 8\end{array}\right|$

Formula used:

If either an entire row or an entire column is 0 then the value of determinant is 0.

Calculation:

Consider the determinant

  $\left|\begin{array}{ccc}2& -1& 1\\ 1& 3& -3\\ 7& -8& 8\end{array}\right|$

Add the corresponding entries of column three to column two.

  $\left|\begin{array}{ccc}2& -1+1& 1\\ 1& 3+\left(-3\right)& -3\\ 7& -8+8& 8\end{array}\right|=\left|\begin{array}{ccc}2& 0& 1\\ 1& 0& -3\\ 7& 0& 8\end{array}\right|$

Observe that all entries of second column are zero.

Recall that if either an entire row or an entire column is 0 then the value of determinant is 0.

Therefore,

  $\begin{array}{c}\left|\begin{array}{ccc}2& -1+1& 1\\ 1& 3+\left(-3\right)& -3\\ 7& -8+8& 8\end{array}\right|=\left|\begin{array}{ccc}2& 0& 1\\ 1& 0& -3\\ 7& 0& 8\end{array}\right|\\ =0\end{array}$

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