Chapter 16, Problem 8CR

To determine

To find: The correct option for the value of given determinant.

Answer to Problem 8CR

The correct option is $\text{a}$ .

Explanation of Solution

Given information:

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

Calculation:

Multiply each element of third row by its minor. Also multiply the product for the element in the third row second column by $-1$ . Since the sum $1+2$ or $3$ .

Add the product.

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

Therefore,the correct option is $\text{a}$ .