Question

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

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

Short answer

The value of the determinant$\left|\begin{array}{ccc}A& B& C\\ 2& 1& 4\\ 1& 3& 1\end{array}\right|$is,$-11A+2B+5C$.

Step-by-step solution

Step 1 Given determinant is,

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

Evaluate the determinant.

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

Step 2 Now simplify it.

$$\begin{gathered} \left|\begin{array}{ccc}A& B& C\\ 2& 1& 4\\ 1& 3& 1\end{array}\right|=A\left[1-12\right]-B\left[2-4\right]+C\left[6-1\right] \\ =A\left[-11\right]-B\left[-2\right]+C\left[5\right] \\ =-11A+2B+5C \end{gathered}$$

Thus, the value of the determinant$\left|\begin{array}{ccc}A& B& C\\ 2& 1& 4\\ 1& 3& 1\end{array}\right|$is,$-11A+2B+5C$.