Question

In Problems $7-14$, find the value of each determinant.

$\left|\begin{array}{ccc}A& B& C\\ 0& 2& 4\\ 3& 1& 3\end{array}\right|$

Short answer

The value of the determinant$\left|\begin{array}{ccc}A& B& C\\ 0& 2& 4\\ 3& 1& 3\end{array}\right|$is,$2A+12B-6C$.

Step-by-step solution

Step 1 Given determinant is,

$\left|\begin{array}{ccc}A& B& C\\ 0& 2& 4\\ 3& 1& 3\end{array}\right|$

Evaluate the determinant.

$\left|\begin{array}{ccc}A& B& C\\ 0& 2& 4\\ 3& 1& 3\end{array}\right|=A\left[2\left(3\right)-4\left(1\right)\right]-B\left[0\left(3\right)-4\left(3\right)\right]+C\left[0\left(1\right)-2\left(3\right)\right]$

Step 2 Now simplify it.

$$\begin{gathered} \left|\begin{array}{ccc}A& B& C\\ 0& 2& 4\\ 3& 1& 3\end{array}\right|=A\left[6-4\right]-B\left[0-12\right]+C\left[0-6\right] \\ =A\left[2\right]-B\left[-12\right]+C\left[-6\right] \\ =2A+12B-6C \end{gathered}$$

Thus, the value of the determinant is,$2A+12B-6C$.