Question

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

$\left|\begin{array}{ccc}A& B& C\\ -1& 3& 5\\ 5& 0& -2\end{array}\right|$

Short answer

The value of the determinant$\left|\begin{array}{ccc}A& B& C\\ -1& 3& 5\\ 5& 0& -2\end{array}\right|$is,$-6A+23B-15C$.

Step-by-step solution

Step 1 Given determinant is,

$\left|\begin{array}{ccc}A& B& C\\ -1& 3& 5\\ 5& 0& -2\end{array}\right|$

Evaluate the determinant.

role="math" localid="1646807619254" $\left|\begin{array}{ccc}A& B& C\\ -1& 3& 5\\ 5& 0& -2\end{array}\right|=A\left[3\left(-2\right)-5\left(0\right)\right]-B\left[-2\left(-1\right)-5\left(5\right)\right]+C\left[-1\left(0\right)-3\left(5\right)\right]$

Step 2 Now simplify it.

$$\begin{gathered} \left|\begin{array}{ccc}A& B& C\\ -1& 3& 5\\ 5& 0& -2\end{array}\right|==A\left[-6-0\right]-B\left[2-25\right]+C\left[-15\right] \\ =A\left[-6\right]-B\left[-23\right]+C\left[-15\right] \\ =-6A+23B-15C \end{gathered}$$

Thus, the value of the determinant is,$-6A+23B-15C$.