Question

Evaluate each determinant using expansion by minors.

$\left|\begin{array}{ccc}3& 7& 6\\ -1& 6& 2\\ 8& -3& -5\end{array}\right|$

Short answer

The value of the determinant $\left|\begin{array}{ccc}3& 7& 6\\ -1& 6& 2\\ 8& -3& -5\end{array}\right|=-265$.

Step-by-step solution

- Decide row to use for expansion

In this case, use first row for the expansion by minors.

$\left|\begin{array}{ccc}3& 7& 6\\ -1& 6& 2\\ 8& -3& -5\end{array}\right|=3\left|\begin{array}{cc}6& 2\\ -3& -5\end{array}\right|-7\left|\begin{array}{cc}-1& 2\\ 8& -5\end{array}\right|+6\left|\begin{array}{cc}-1& 6\\ 8& -3\end{array}\right|$

- Evaluate 2×2 determinants

The determinant of second order matrix is found by calculating the difference of the product of the two diagonals, that is., $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$. Apply this definition to find the $2\times 2$determinant.

$\begin{array}{c}\left|\begin{array}{ccc}3& 7& 6\\ -1& 6& 2\\ 8& -3& -5\end{array}\right|=3\left|\begin{array}{cc}6& 2\\ -3& -5\end{array}\right|-7\left|\begin{array}{cc}-1& 2\\ 8& -5\end{array}\right|+6\left|\begin{array}{cc}-1& 6\\ 8& -3\end{array}\right|\\ =3\left(-30-\left(-6\right)\right)-7\left(5-16\right)+6\left(3-48\right)\end{array}$

- Multiply and simplify

Further simplify the values.

$\begin{array}{c}\left|\begin{array}{ccc}3& 7& 6\\ -1& 6& 2\\ 8& -3& -5\end{array}\right|=3\left(-30-\left(-6\right)\right)-7\left(5-16\right)+6\left(3-48\right)\\ =3\left(-24\right)-7\left(-11\right)+6\left(-45\right)\\ =-72+77-270\\ =-265\end{array}$