Chapter 16.8, Problem 1OE

To determine

To calculate: The value of the determinant,

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

Answer to Problem 1OE

The value of the determinant is $-1$ .

Explanation of Solution

Given information:

The value of the determinant:

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

Formula used:

The sign of the determinant changes when any two rows or any two columns of a determinant are interchanged.

Calculation:

Consider the determinant

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

It is provided that the value of the determinant

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

Observe that the first and the second row of the determinant are interchanged.

The sign of the determinant changes when any two rows or any two columns of a determinant are interchanged.

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

Thus, the value of the determinant is $-1$ .