Question

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

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

Short answer

The value of the determinant$\left|\begin{array}{ccc}A& B& C\\ 1& -2& -3\\ 0& 2& -2\end{array}\right|$is,role="math" localid="1646808146748" $10A+2B+2C$.

Step-by-step solution

Step 1 Given determinant is,

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

Evaluate the determinant.

role="math" localid="1646808166729" $\left|\begin{array}{ccc}A& B& C\\ 1& -2& -3\\ 0& 2& -2\end{array}\right|=A\left[-2\left(-2\right)-\left(-3\right)\left(2\right)\right]-B\left[1\left(-2\right)-0\left(-3\right)\right]+C\left[1\left(2\right)-0(-2)\right]$

Step 2 Now simplify it.

$$\begin{gathered} \left|\begin{array}{ccc}A& B& C\\ 1& -2& -3\\ 0& 2& -2\end{array}\right|=A\left[4+6\right]-B\left[-2-0\right]+C\left[2-0\right] \\ =A\left[10\right]-B\left[-2\right]+C\left[2\right] \\ =10A+2B+2C \end{gathered}$$

Thus the value of determinant is,$10A+2B+2C$.