Question

Evaluate each determinant by expanding by minors.

$\left|\begin{array}{ccc}3& 5& 4\\ -1& 3& 0\\ -2& 6& 1\end{array}\right|$

Short answer

The required value of the determinant is$14$

Step-by-step solution

Step 1. Given information

The determinant is$\left|\begin{array}{ccc}3& 5& 4\\ -1& 3& 0\\ -2& 6& 1\end{array}\right|$

Step 2. Expand by minors along the second row to evaluate determinant

$$\begin{gathered} \left|\begin{array}{ccc}3& 5& 4\\ -1& 3& 0\\ -2& 6& 1\end{array}\right| \\ =-(-1)\left|\begin{array}{cc}5& 4\\ 6& 1\end{array}\right|+3\left|\begin{array}{cc}3& 4\\ -2& 1\end{array}\right|-0\left|\begin{array}{cc}3& 5\\ -2& 6\end{array}\right| \\ =1\left[\right(5\left)\right(1)-(4\left)\right(6\left)\right]+3\left[\right(3\left)\right(1)-(4\left)\right(-2\left)\right]-0 \\ =1[-19]+3\left[11\right]-0 \\ =-19+33-0 \\ =14 \end{gathered}$$

Step 3. Required answer

The required value of the determinant is$14$