Chapter 16, Problem 9CR

To determine

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

Answer to Problem 9CR

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

Explanation of Solution

Given information:

The given determinant is

  $\left|\begin{array}{ccc}3& 2& -1\\ 1& 0& 1\\ -12& 3& 4\end{array}\right|$ .

Calculation:

Multiply each element in the first column by $-1$ . Add the resulting product to the corresponding elements in the third column.

  $\begin{array}{c}2\left|\begin{array}{ccc}3& 2& -1\\ 1& 0& 1\\ -12& 3& 4\end{array}\right|=2\left|\begin{array}{ccc}3& 2& -1+3\left(-1\right)\\ 1& 0& 1+1\left(-1\right)\\ -12& 3& 4+\left(-12\right)\left(-1\right)\end{array}\right|\\ =2\left|\begin{array}{ccc}3& 2& -4\\ 1& 0& 0\\ -12& 3& 16\end{array}\right|\end{array}$

Factor out $4$ from the elements in the third column.

  $\begin{array}{c}2\left|\begin{array}{ccc}3& 2& -4\\ 1& 0& 0\\ -12& 3& 16\end{array}\right|=2\left(4\right)\left|\begin{array}{ccc}3& 2& -1\\ 1& 0& 0\\ -12& 3& 4\end{array}\right|\\ =8\left|\begin{array}{ccc}3& 2& -1\\ 1& 0& 0\\ -12& 3& 4\end{array}\right|\end{array}$

Calculate the value by expanding the determinant about row two.

  $\begin{array}{c}8\left|\begin{array}{ccc}3& 2& -1\\ 1& 0& 0\\ -12& 3& 4\end{array}\right|=8\left(\left(-1\right)\left(1\right)\left|\begin{array}{cc}2& -1\\ 3& 4\end{array}\right|+0+0\right)\\ =8\left[-1\left(8+3\right)\right]\\ =8\left(-11\right)\\ =-88\end{array}$

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